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 three given vectors. If $\overrightarrow r$ is a vector such that $$\overrightarrow r \times \overrightarrow a = \overrightarrow c \times \overrightarrow a $$ and $\overrightarrow r .\,\overrightarrow b = 0$, then $\overrightarrow r .\,\overrightarrow a$ is equal to __________.
Answer (integer)
12
Solution
<p>Given, $\overrightarrow a = \widehat i + 2\widehat j - \widehat k$,</p>
<p>$\overrightarrow b = \widehat i - \widehat j$,</p>
<p>$\overrightarrow c = \widehat i - \widehat j - \widehat k$</p>
<p>$$\overrightarrow r \times \overrightarrow a = \overrightarrow c \times \overrightarrow a $$</p>
<p>$$ \Rightarrow \overrightarrow r \times \overrightarrow a - \overrightarrow c \times \overrightarrow a = 0$$</p>
<p>$$ \Rightarrow (\overrightarrow r - \overrightarrow c ) \times \overrightarrow a = 0$$</p>
<p>$\therefore$ $\overrightarrow r - \overrightarrow c = \lambda \overrightarrow a$</p>
<p>$\Rightarrow \overrightarrow r = \lambda \overrightarrow a + \overrightarrow c$</p>
<p>$$ \Rightarrow \overrightarrow r \,.\,\overrightarrow b = \lambda \overrightarrow a \,.\,\overrightarrow b + \overrightarrow c \,.\,\overrightarrow b $$ (taking dot with $\overrightarrow b$)</p>
<p>$$ \Rightarrow 0 = \lambda \overrightarrow a \,.\,\overrightarrow b + \overrightarrow c \,.\,\overrightarrow b $$ [$\because$ $\overrightarrow r \,.\,\overrightarrow b = 0$]</p>
<p>$$ \Rightarrow \lambda (\widehat i + 2\widehat j - \widehat k)\,.\,(\widehat i - \widehat j) + (\widehat i - \widehat j - \widehat k)\,.\,(\widehat i - \widehat j) = 0$$</p>
<p>$\Rightarrow \lambda (1 - 2) + 2 = 0$</p>
<p>$\Rightarrow \lambda = 2$</p>
<p>$\therefore$ $\overrightarrow r = 2\overrightarrow a + \overrightarrow c$</p>
<p>$$ \Rightarrow \overrightarrow r \,.\,\overrightarrow a = 2\overrightarrow a \,.\,\overrightarrow a + \overrightarrow c \,.\,\overrightarrow a $$ [taking dot with ${\overrightarrow a }$]</p>
<p>$$ = 2{\left| {\overrightarrow a } \right|^2} + \overrightarrow a \,.\,\overrightarrow c $$</p>
<p>$= 2(1 + 4 + 1) + (1 - 2 + 1)$</p>
<p>$\Rightarrow \overrightarrow r \,.\,\overrightarrow a = 12$</p>
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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