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 $$\overrightarrow a .\,\overrightarrow c = \left| {\overrightarrow c } \right|,\left| {\overrightarrow c - \overrightarrow a } \right| = 2\sqrt 2 $$ and the angle between $(\overrightarrow a \times \overrightarrow b )$ and $\overrightarrow c$ is ${\pi \over 6}$, then the value of $$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c } \right|$$ is :
Solution
$$\left| {\overrightarrow a } \right| = 3 = a;\overrightarrow a \,.\,\overrightarrow c = c$$<br><br>Now, $\left| {\overrightarrow c - \overrightarrow a } \right| = 2\sqrt 2$<br><br>$\Rightarrow {c^2} + {a^2} - 2\overrightarrow c \,.\,\overrightarrow a = 8$<br><br>$\Rightarrow {c^2} + 9 - 2(c) = 8$<br><br>$$ \Rightarrow {c^2} - 2c + 1 = 0 \Rightarrow c = 1 = \left| {\overrightarrow c } \right|$$<br><br>Also, $$\overrightarrow a \times \overrightarrow b = 2\widehat i - 2\widehat j + \widehat k$$<br><br>Given, $$(\overrightarrow a \times \overrightarrow b ) = \left| {\overrightarrow a \times \overrightarrow b } \right|\left| {\overrightarrow c } \right|\sin {\pi \over 6}$$<br><br>$= (3)(1)(1/2)$<br><br>$= 3/2$
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
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