JEE MAIN · MATHEMATICS · VECTOR ALGEBRA
JEE Main Vector Algebra Questions & Solutions
169 solved questions on Vector Algebra, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
169 solved questions on Vector Algebra, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3$ and $|\vec{a} \times \vec{b}|^{2}=75$. Then…
View solution →If $\left( {\overrightarrow a + 3\overrightarrow b } \right)$ is perpendicular to $\left( {7\overrightarrow a - 5\overrightarrow b } \right)$ and $\left( {\overrightarrow a -…
View solution →The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha \hat{i}+2 \alpha \hat{j}-2 \hat{k}$ is…
View solution →Let $\overrightarrow a$ = $\widehat i$ + 2$\widehat j$ $-$ 3$\widehat k$ and $\overrightarrow b = 2\widehat i$ $-$ 3$\widehat j$ + 5$\widehat k$. If $\overrightarrow r$ $\times$…
View solution →Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$, and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$ and the angle…
View solution →Let $\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ and let $\vec{b}$ be a vector such that $\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$ and $\vec{a} \cdot \vec{b}=3$. Then the projection of…
View solution →Let three vectors $\overrightarrow a$, $\overrightarrow b$ and $\overrightarrow c$ be such that $\overrightarrow a \times \overrightarrow b = \overrightarrow c$, $\overrightarrow…
View solution →Let $\overrightarrow a = \alpha \widehat i + 2\widehat j - \widehat k$ and $\overrightarrow b = - 2\widehat i + \alpha \widehat j + \widehat k$, where $\alpha \in R$. If the area…
View solution →Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=\sqrt{14},|\vec{b}|=\sqrt{6}$ and $|\vec{a} \times \vec{b}|=\sqrt{48}$. Then $(\vec{a} \cdot \vec{b})^{2}$ is equal…
View solution →Let $\overrightarrow a$, $\overrightarrow b$ and $\overrightarrow c$ be three vectors such that $\left| {\overrightarrow a } \right| = \sqrt 3$, $$\left| {\overrightarrow b }…
View solution →Let $\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$. If $\overrightarrow{\mathrm{c}}$ is a is vector such that…
View solution →Let $\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b} \cdot…
View solution →Let O be the origin. Let $\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k$ and $\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$, x, y$\in$R, x > 0,…
View solution →A vector $$\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)$$ lies in the plane of the vectors, $\overrightarrow b =…
View solution →Let three vectors $\overrightarrow a ,\overrightarrow b$ and $\overrightarrow c$ be such that $\overrightarrow c$ is coplanar with $\overrightarrow a$ and $\overrightarrow b$,…
View solution →Let $\overrightarrow a$ and $\overrightarrow b$ be two vectors such that $$\left| {2\overrightarrow a + 3\overrightarrow b } \right| = \left| {3\overrightarrow a + \overrightarrow…
View solution →Let a vector ${\overrightarrow a }$ be coplanar with vectors $\overrightarrow b = 2\widehat i + \widehat j + \widehat k$ and $\overrightarrow c = \widehat i - \widehat j +…
View solution →$$ \text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b}) \times \hat{i}) \cdot…
View solution →Let $\overrightarrow a$ , $\overrightarrow b$ and $\overrightarrow c$ be three unit vectors such that $\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0$. If…
View solution →If $$\overrightarrow a \,.\,\overrightarrow b = 1,\,\overrightarrow b \,.\,\overrightarrow c = 2$$ and $\overrightarrow c \,.\,\overrightarrow a = 3$, then the value of $$\left[…
View solution →Let $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ three vectors mutually perpendicular to each other and have same magnitude. If a vector ${ \overrightarrow r }$…
View solution →Let $$\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}$$ and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha0$. If the minimum value of the…
View solution →Let $\vec{a}=3 \hat{i}+2 \hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}) \times \vec{c}=2(\vec{a} \times \vec{b})+24…
View solution →If vectors $\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k$ and $\overrightarrow {{a_2}} = \widehat i + y\widehat j + z\widehat k$ are collinear, then a possible…
View solution →Let the vectors $\vec{u}_{1}=\hat{i}+\hat{j}+a \hat{k}, \vec{u}_{2}=\hat{i}+b \hat{j}+\hat{k}$ and $\vec{u}_{3}=c \hat{i}+\hat{j}+\hat{k}$ be coplanar. If the vectors…
View solution →Let $a, b, c$ be three distinct real numbers, none equal to one. If the vectors $$a \hat{i}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b \hat{j}+\hat{\mathrm{k}}$$ and…
View solution →Let $\vec{a}$ and $\vec{b}$ be two vectors, Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$, then the value of…
View solution →Let $\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha0$. If the projection of…
View solution →If four distinct points with position vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are coplanar, then $[\vec{a} \,\,\vec{b} \,\,\vec{c}]$ is equal to :
View solution →Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5 \vec{b}$ is collinear with $\vec{c}, \vec{b}+6…
View solution →If the shortest distance between the lines $$\overrightarrow {{r_1}} = \alpha \widehat i + 2\widehat j + 2\widehat k + \lambda (\widehat i - 2\widehat j + 2\widehat k)$$,…
View solution →The vector $\overrightarrow a = - \widehat i + 2\widehat j + \widehat k$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is…
View solution →Let $|\vec{a}|=2,|\vec{b}|=3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{\pi}{4}$. Then $|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^{2}$ is equal…
View solution →Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c}=11, \vec{b} \cdot(\vec{a} \times…
View solution →Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to both…
View solution →Let $\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$ and $\overrightarrow b = \widehat i + 2\widehat j - \widehat k$. Let a vector $\overrightarrow v$ be in the plane…
View solution →Let x0 be the point of Local maxima of $$f(x) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)$$, where $\overrightarrow a = x\widehat i -…
View solution →If the points with position vectors $$\alpha \hat{i}+10 \hat{j}+13 \hat{k}, 6 \hat{i}+11 \hat{j}+11 \hat{k}, \frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}$$ are collinear, then $(19…
View solution →Let $$\vec{a}=2 \hat{i}+\hat{j}-\hat{k}, \vec{b}=((\vec{a} \times(\hat{i}+\hat{j})) \times \hat{i}) \times \hat{i}$$. Then the square of the projection of $\vec{a}$ on $\vec{b}$…
View solution →Let $$\overrightarrow a = \widehat i + 2\widehat j + \lambda \widehat k,\overrightarrow b = 3\widehat i - 5\widehat j - \lambda \widehat k,\overrightarrow a \,.\,\overrightarrow c…
View solution →Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}$ be three vectors. If $\vec{r}$ is a vector such that,…
View solution →Let $$\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}$$ and $\vec{c}$ be vectors such that $\vec{a} \times \vec{c}=\vec{a} \times \vec{b}$. If…
View solution →Let the vectors$$(2 + a + b)\widehat i + (a + 2b + c)\widehat j - (b + c)\widehat k,(1 + b)\widehat i + 2b\widehat j - b\widehat k$$ and $(2 + b)\widehat i + 2b\widehat j + (1 -…
View solution →Let $\overrightarrow a$, $\overrightarrow b$, $\overrightarrow c$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the…
View solution →Let $$\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k}$$ and $\overrightarrow{\mathrm{c}}$ be a vector such that…
View solution →Let $\lambda \in \mathbb{R}, \vec{a}=\lambda \hat{i}+2 \hat{j}-3 \hat{k}, \vec{b}=\hat{i}-\lambda \hat{j}+2 \hat{k}$. If $((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b}))…
View solution →Let $\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}$ and a vector $\vec{c}$ be such that $$\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times…
View solution →Let $\overrightarrow a$, $\overrightarrow b$ and $\overrightarrow c$ be three vectors such that $\overrightarrow a$ = $\overrightarrow b$ $\times$ ($\overrightarrow b$ $\times$…
View solution →Let the vectors $\vec{a}, \vec{b}, \vec{c}$ represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are…
View solution →Let $$\overrightarrow u = \widehat i - \widehat j - 2\widehat k,\overrightarrow v = 2\widehat i + \widehat j - \widehat k,\overrightarrow v .\,\overrightarrow w = 2$$ and…
View solution →For $\lambda0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{i}+\lambda \hat{j}-3 \hat{k}$ and $\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$. If the vectors…
View solution →Let $\vec{a}=2 \hat{i}+5 \hat{j}-\hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}+2 \hat{k}$ and $\vec{c}$ be three vectors such that $$(\vec{c}+\hat{i})…
View solution →Let $\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}=\vec{b} \times \vec{c}$ and $|\vec{b}|^{2}=50$,…
View solution →Let $\overrightarrow a = 4\widehat i + 3\widehat j$ and $\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k$. If $\overrightarrow c$ is a vector such that…
View solution →Let $\overrightarrow a$, $\overrightarrow b$ and $\overrightarrow c$ be three unit vectors such that ${\left| {\overrightarrow a - \overrightarrow b } \right|^2}$ + ${\left|…
View solution →Let A, B, C be three points whose position vectors respectively are $\overrightarrow a = \widehat i + 4\widehat j + 3\widehat k$ $$\overrightarrow b = 2\widehat i + \alpha…
View solution →Let $\overrightarrow c$ be a vector perpendicular to the vectors, $\overrightarrow a$ = $\widehat i$ + $\widehat j$ $-$ $\widehat k$ and $\overrightarrow b$ = $\widehat i$ +…
View solution →Let $\mathrm{ABCD}$ be a quadrilateral. If $\mathrm{E}$ and $\mathrm{F}$ are the mid points of the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ respectively and $(\overrightarrow{A…
View solution →The lines $$\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)$$ and $$\overrightarrow r = \left( {2\widehat i - \widehat j}…
View solution →Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a}+2 \vec{b}$ and $3 \vec{a}-\lambda \vec{b}$ are…
View solution →Let $\overrightarrow a$, $\overrightarrow b$, $\overrightarrow c$ be three non-coplanar vectors such that $\overrightarrow a$ $\times$ $\overrightarrow b$ = 4$\overrightarrow c$,…
View solution →Let $\widehat a$, $\widehat b$ be unit vectors. If $\overrightarrow c$ be a vector such that the angle between $\widehat a$ and $\overrightarrow c$ is ${\pi \over {12}}$, and…
View solution →Let $\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b$ and $\overrightarrow c = \widehat j - \widehat k$ be three vectors such that $\overrightarrow a…
View solution →Consider three vectors $\vec{a}, \vec{b}, \vec{c}$. Let $|\vec{a}|=2,|\vec{b}|=3$ and $\vec{a}=\vec{b} \times \vec{c}$. If $\alpha \in\left[0, \frac{\pi}{3}\right]$ is the angle…
View solution →Let $\theta$ be the angle between the vectors $\overrightarrow a$ and $\overrightarrow b$, where $|\overrightarrow a | = 4,$ $|\overrightarrow b | = 3$ and $\theta \in \left(…
View solution →Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j}, \hat{i}+\hat{k}$ and $\hat{i}-\hat{j}, \hat{j}-\hat{k}$.…
View solution →Let $\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$ be two vectors. Then which one of the following statements is TRUE ?
View solution →Let $$\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+3 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}+3 \hat{j}-5 \hat{k}$$ and $\overrightarrow{\mathrm{c}}=3…
View solution →Let $\overrightarrow a = \widehat i - \alpha \widehat j + \beta \widehat k$, $\overrightarrow b = 3\widehat i + \beta \widehat j - \alpha \widehat k$ and $\overrightarrow c =…
View solution →Let $$\overrightarrow{\mathrm{a}}=4 \hat{i}-\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=11 \hat{i}-\hat{j}+\hat{k}$$ and $\overrightarrow{\mathrm{c}}$ be a vector such that…
View solution →If $\overrightarrow a$ and $\overrightarrow b$ are perpendicular, then $$\overrightarrow a \times \left( {\overrightarrow a \times \left( {\overrightarrow a \times \left(…
View solution →Let $\overrightarrow a$, $\overrightarrow b$ and $\overrightarrow c$ be three non zero vectors such that $\overrightarrow b$ . $\overrightarrow c$ = 0 and $$\overrightarrow a…
View solution →Let $\overrightarrow a$ be a vector which is perpendicular to the vector $3\widehat i + {1 \over 2}\widehat j + 2\widehat k$. If $$\overrightarrow a \times \left( {2\widehat i +…
View solution →Let $\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$, $\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$ and $\overrightarrow c = \widehat i +…
View solution →Let $\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}$ and $\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}$ be three given vectors. If $\overrightarrow{\mathrm{r}}$ is a vector…
View solution →Let O be the origin and the position vector of the point P be $- \widehat i - 2\widehat j + 3\widehat k$. If the position vectors of the points A, B and C are $- 2\widehat i +…
View solution →Let $\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$ and $\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}$ be two vectors, such that $\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}$.…
View solution →Let $\overrightarrow a = 2\widehat i + \widehat j - 2\widehat k$ and $\overrightarrow b = \widehat i + \widehat j$. If $\overrightarrow c$ is a vector such that $$\overrightarrow…
View solution →Let $\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k$ and $\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k$. If the area of the parallelogram…
View solution →Let $\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$…
View solution →Let the vectors $$\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}, \vec{b}=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}$$ and $\vec{c}=t \hat{i}-t \hat{j}+\hat{k}, t \in \mathbf{R}$ be such…
View solution →Let $$\overrightarrow a = - \widehat i - \widehat j + \widehat k,\overrightarrow a \,.\,\overrightarrow b = 1$$ and $\overrightarrow a \times \overrightarrow b = \widehat i -…
View solution →If the projection of the vector $\widehat i + 2\widehat j + \widehat k$ on the sum of the two vectors $2\widehat i + 4\widehat j - 5\widehat k$ and $- \lambda \widehat i +…
View solution →For any vector $\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$, with $10\left|a_{i}\right| (A): $$\max \left\{\left|a_{1}\right|,\left|a_{2}\right|,\left|a_{3}\right|\right\}…
View solution →Let a, b c $\in$ R be such that a2 + b2 + c2 = 1. If $$a\cos \theta = b\cos \left( {\theta + {{2\pi } \over 3}} \right) = c\cos \left( {\theta + {{4\pi } \over 3}} \right)$$,…
View solution →Let $\overrightarrow b = \widehat i + \widehat j + \lambda \widehat k$, $\lambda$ $\in$ R. If $\overrightarrow a$ is a vector such that $$\overrightarrow a \times \overrightarrow…
View solution →If $\overrightarrow a = 2\widehat i + \widehat j + 3\widehat k$, $\overrightarrow b = 3\widehat i + 3\widehat j + \widehat k$ and $\overrightarrow c = {c_1}\widehat i +…
View solution →Let $\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}$ and $\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}$ be three given vectors. If $\vec{r}$ is a vector…
View solution →Let a unit vector $\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}$ make angles $\frac{\pi}{2}, \frac{\pi}{3}$ and $\frac{2 \pi}{3}$ with the vectors $$\frac{1}{\sqrt{2}}…
View solution →Let $\mathrm{ABC}$ be a triangle of area $15 \sqrt{2}$ and the vectors $$\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=\mathrm{a}…
View solution →Let $\overrightarrow a$ and $\overrightarrow b$ be two non-zero vectors perpendicular to each other and $|\overrightarrow a | = |\overrightarrow b |$. If $|\overrightarrow a…
View solution →Let a vector $\vec{a}$ has magnitude 9. Let a vector $\vec{b}$ be such that for every $(x, y) \in \mathbf{R} \times \mathbf{R}-\{(0,0)\}$, the vector $(x \vec{a}+y \vec{b})$ is…
View solution →Let a vector $\overrightarrow c$ be coplanar with the vectors $\overrightarrow a = - \widehat i + \widehat j + \widehat k$ and $\overrightarrow b = 2\widehat i + \widehat j -…
View solution →Let the position vectors of the points A, B, C and D be $$5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}$$ and $-\hat{i}+5…
View solution →Let a, b and c be distinct positive numbers. If the vectors $a\widehat i + a\widehat j + c\widehat k,\widehat i+\widehat k$ and $c\widehat i + c\widehat j + b\widehat k$ are…
View solution →Let $\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in \mathbb{R}$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$…
View solution →The sum of all values of $\alpha$, for which the points whose position vectors are $$\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}-3 \hat{j}+4 \hat{k},(\alpha+1) \hat{i}+2 \hat{k}$$ and…
View solution →A vector $\overrightarrow a$ has components 3p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter…
View solution →Between the following two statements: Statement I : Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying…
View solution →Let S be the set of all a $\in R$ for which the angle between the vectors $\vec{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}$ and $$\vec{v}=\left(\log _{e} b\right)…
View solution →If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\widehat j + 4\widehat…
View solution →If $\overrightarrow a$ and $\overrightarrow b$ are unit vectors, then the greatest value of $$\sqrt 3 \left| {\overrightarrow a + \overrightarrow b } \right| + \left|…
View solution →If $\overrightarrow a = \alpha \widehat i + \beta \widehat j + 3\widehat k$,$\overrightarrow b = - \beta \widehat i - \alpha \widehat j - \widehat k$ and $\overrightarrow c =…
View solution →Let $\lambda \in \mathbb{Z}, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$. Let $\vec{c}$ be a vector such that $$(\vec{a}+\vec{b}+\vec{c})…
View solution →Let a unit vector $\widehat{O P}$ make angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes $\mathrm{OX}$, $\mathrm{OY}, \mathrm{OZ}$ respectively,…
View solution →Let $\vec{a}, \vec{b}, \vec{c}$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $$(\vec{a}…
View solution →Let $\overrightarrow a$ and $\overrightarrow b$ be the vectors along the diagonals of a parallelogram having area $2\sqrt 2$. Let the angle between $\overrightarrow a$ and…
View solution →Let $\overrightarrow a$, $\overrightarrow b$ and $\overrightarrow c$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A,B,C$ and $D$ be…
View solution →If $\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k$, then the value of $${\left| {\widehat i \times \left( {\overrightarrow a \times \widehat i} \right)} \right|^2} +…
View solution →Let $\overrightarrow a = \widehat i - 2\widehat j + \widehat k$ and $\overrightarrow b = \widehat i - \widehat j + \widehat k$ be two vectors. If $\overrightarrow c$ is a vector…
View solution →Let $\overrightarrow a = \widehat i + 2\widehat j - \widehat k$, $\overrightarrow b = \widehat i - \widehat j$ and $\overrightarrow c = \widehat i - \widehat j - \widehat k$ be…
View solution →Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{i}+2 \hat{j}-\hat{k}$ and an angle of $45^{\circ}$ with $\hat{i}-\hat{k}$ be $\vec{C}$. Then…
View solution →Let $\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k$, $\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k$ and $\overrightarrow c = - \widehat i +…
View solution →Let $$\vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{k}, \vec{c}=\beta \hat{j}-\hat{k}$$, where $\alpha$ and $\beta$ are integers and $\alpha \beta=-6$. Let the…
View solution →Let $\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}$ and $\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}$ be three vectors. If a vectors $\vec{p}$ satisfies…
View solution →Let $\overrightarrow{\mathrm{a}}=2 \hat{i}-3 \hat{j}+\hat{k}, \quad \overrightarrow{\mathrm{~b}}=3 \hat{i}+2 \hat{j}+5 \hat{k}$ and a vector $\overrightarrow{\mathrm{c}}$ be such…
View solution →Let $\overrightarrow a = \widehat i + \widehat j - \widehat k$ and $\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$. Then the number of vectors $\overrightarrow b$…
View solution →Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d} \times…
View solution →If the vectors $\overrightarrow a = \lambda \widehat i + \mu \widehat j + 4\widehat k$, $\overrightarrow b = - 2\widehat i + 4\widehat j - 2\widehat k$ and $\overrightarrow c =…
View solution →If the vectors, $$\overrightarrow p = \left( {a + 1} \right)\widehat i + a\widehat j + a\widehat k$$, $$\overrightarrow q = a\widehat i + \left( {a + 1} \right)\widehat j +…
View solution →Let the vectors $\overrightarrow a$, $\overrightarrow b$, $\overrightarrow c$ be such that $\left| {\overrightarrow a } \right| = 2$, $\left| {\overrightarrow b } \right| = 4$ and…
View solution →If $\overrightarrow x$ and $\overrightarrow y$ be two non-zero vectors such that $$\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|$$…
View solution →Let $\overrightarrow{\mathrm{a}}=3 \hat{i}+\hat{j}$ and $\overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}+\hat{k}$. Let $\overrightarrow{\mathrm{c}}$ be a vector satisfying…
View solution →Let $\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k$ and $\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k$. Let ${\overrightarrow \beta _1}$ be…
View solution →Let $\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$ and $\overrightarrow q = \widehat i + 2\widehat j + \widehat k$ be two vectors. If a vector $\overrightarrow r =…
View solution →Let $\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$, $\overrightarrow b = \widehat i + \widehat j + \widehat k$ and $\overrightarrow c$ be a vector such that…
View solution →If $\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5$ and $\left| {\overrightarrow a \times \overrightarrow b } \right|$ = 8, then $\left|…
View solution →Let $\overrightarrow x$ be a vector in the plane containing vectors $\overrightarrow a = 2\widehat i - \widehat j + \widehat k$ and $\overrightarrow b = \widehat i + 2\widehat j -…
View solution →Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{i}-\hat{j}+\hat{k}, \hat{i}+2 \hat{j}+\mu \hat{k}$ and $3 \hat{i}-4 \hat{j}+5 \hat{k}$, where…
View solution →Let $\widehat a$ and $\widehat b$ be two unit vectors such that $|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2$. If $\theta$ $\in$ (0, $\pi$) is the angle…
View solution →Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and…
View solution →Let $\overrightarrow a$ = 2$\widehat i$ $-$ 3$\widehat j$ + 4$\widehat k$ and $\overrightarrow b$ = 7$\widehat i$ + $\widehat j$ $-$ 6$\widehat k$.If $\overrightarrow r$ $\times$…
View solution →If $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ are three non-zero vectors and $\widehat n$ is a unit vector perpendicular to $\overrightarrow c$ such that…
View solution →The area of the quadrilateral $\mathrm{ABCD}$ with vertices $\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)$ and $\mathrm{D}(1,-6,-7)$ is equal to :
View solution →Let the volume of a parallelopiped whose coterminous edges are given by $\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k$, $\overrightarrow v = \widehat i +…
View solution →Let $\overrightarrow a = \widehat i + \widehat j + \widehat k$ and $\overrightarrow b = \widehat j - \widehat k$. If $\overrightarrow c$ is a vector such that $\overrightarrow a…
View solution →If the four points, whose position vectors are $$3\widehat i - 4\widehat j + 2\widehat k,\widehat i + 2\widehat j - \widehat k, - 2\widehat i - \widehat j + 3\widehat k$$ and…
View solution →$A(2,6,2), B(-4,0, \lambda), C(2,3,-1)$ and $D(4,5,0),|\lambda| \leq 5$ are the vertices of a quadrilateral $A B C D$. If its area is 18 square units, then $5-6 \lambda$ is equal…
View solution →Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b} \times \vec{a}|=2$. Then $|(\vec{b} \times \vec{a})-\vec{b}|^2$ is equal to
View solution →The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$ are inclined at an obtuse angle…
View solution →Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\vec{c}=x \hat{i}+2 \hat{j}+3 \hat{k}, x \in \mathbb{R}$. If $\vec{d}$ is the unit vector in the…
View solution →If the volume of a parallelopiped, whose coterminus edges are given by the vectors $\overrightarrow a = \widehat i + \widehat j + n\widehat k$, $\overrightarrow b = 2\widehat i +…
View solution →If the points $\mathrm{P}$ and $\mathrm{Q}$ are respectively the circumcenter and the orthocentre of a $\triangle \mathrm{ABC}$, then…
View solution →Let $$\overrightarrow{\mathrm{a}}=\mathrm{a}_1 \hat{i}+\mathrm{a}_2 \hat{j}+\mathrm{a}_3 \hat{k}$$ and $$\overrightarrow{\mathrm{b}}=\mathrm{b}_1 \hat{i}+\mathrm{b}_2…
View solution →Let three vectors ,$$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x…
View solution →Let $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$ ${a_i} 0$, $i = 1,2,3$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ.…
View solution →Let $\overrightarrow a = \widehat i + \widehat j + 2\widehat k$ and $\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$. Then the vector product $$\left(…
View solution →Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31}, 4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a} \times \vec{b})=3(\vec{c} \times \vec{a})$. If the angle…
View solution →Let $\overrightarrow{\mathrm{a}}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{\mathrm{b}}=-\hat{i}-8 \hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{c}}=4 \hat{i}+\mathrm{c}_2…
View solution →Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda0$, on the vector $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the…
View solution →Let $\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and…
View solution →Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}, \vec{c}=\lambda \hat{j}+\mu \hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a} \times…
View solution →Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=3 \hat{i}-3 \hat{j}+3 \hat{k}, \vec{c}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{d}$ be a vector such that $\vec{b} \times…
View solution →Let the three sides of a triangle ABC be given by the vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$. Let $G$ be the centroid…
View solution →Let $\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+\hat{k}, $ $\overrightarrow{\mathrm{b}}=3(\hat{i}-\hat{j}+\hat{k})$. Let $\overrightarrow{\mathrm{c}}$ be the vector such that…
View solution →Let $\overrightarrow{\mathrm{a}}=-5 \hat{i}+\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}-4 \hat{k}$ and…
View solution →Let the point A divide the line segment joining the points $\mathrm{P}(-1,-1,2)$ and $\mathrm{Q}(5,5,10)$ internally in the ratio $r: 1(r0)$. If O is the origin and…
View solution →Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the…
View solution →Let $\overrightarrow{\mathrm{a}}=3 \hat{i}-\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\overrightarrow{\mathrm{a}} \times(\hat{i}-2 \hat{k})$ and…
View solution →Let $A, B, C$ be three points in xy-plane, whose position vector are given by $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $a \hat{i}+(1-a) \hat{j}$ respectively with…
View solution →If the components of $\vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ along and perpendicular to $\vec{b}=3 \hat{i}+\hat{j}-\hat{k}$ respectively, are $\frac{16}{11}(3…
View solution →Let $ \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \ \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} $ and $ \vec{c} $ be a vector such that $ \vec{a} \times \vec{c} = \vec{a} \times \vec{b}…
View solution →Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{b}=2 \hat{i}+7 \hat{j}+3 \hat{k}$. Let $\mathrm{L}_1 : \overrightarrow{\mathrm{r}}=(-\hat{i}+2 \hat{j}+\hat{k})+\lambda \vec{a},…
View solution →Let $ \hat{a} $ be a unit vector perpendicular to the vectors $ \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} $ and $ \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} $, and $ \hat{a} $ makes an…
View solution →If $\overrightarrow{\mathrm{a}}$ is a nonzero vector such that its projections on the vectors $2 \hat{i}-\hat{j}+2 \hat{k}, \hat{i}+2 \hat{j}-2 \hat{k}$ and $\hat{k}$ are equal,…
View solution →Consider two vectors $\vec{u}=3 \hat{i}-\hat{j}$ and $\vec{v}=2 \hat{i}+\hat{j}-\lambda \hat{k}, \lambda0$. The angle between them is given by $\cos ^{-1}\left(\frac{\sqrt{5}}{2…
View solution →Let the angle $\theta, 0
View solution →Let $ \vec{a} $ and $ \vec{b} $ be the vectors of the same magnitude such that $ \frac{|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|}{|\vec{a} + \vec{b}| - |\vec{a} - \vec{b}|} =…
View solution →Let $ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} $ and $ \vec{b} = 2\hat{i} + \hat{j} - \hat{k} $. Let $ \hat{c} $ be a unit vector in the plane of the vectors $ \vec{a} $ and $…
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