JEE MAIN - Mathematics (2021 - 27th July Morning Shift - No. 18)
Let $$\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b $$ and $$\overrightarrow c = \widehat j - \widehat k$$ be three vectors such that $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$ and $$\overrightarrow a \,.\,\overrightarrow b = 1$$. If the length of projection vector of the vector $$\overrightarrow b $$ on the vector $$\overrightarrow a \times \overrightarrow c $$ is l, then the value of 3l2 is equal to _____________.
Answer
2
Explanation
$$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$
Take Dot with $$\overrightarrow c $$
$$\left( {\overrightarrow a \times \overrightarrow b } \right).\,\overrightarrow c = {\left| {\overrightarrow c } \right|^2} = 2$$
Projection of $$\overrightarrow b $$ or $$\overrightarrow a \times \overrightarrow c = l$$
$${{\left| {\overrightarrow b \,.\,(\overrightarrow a \times \overrightarrow c )} \right|} \over {|\overrightarrow a \times \overrightarrow c |}} = l$$
$$\therefore$$ $$l = {2 \over {\sqrt 6 }} \Rightarrow {l^2} = {4 \over 6}$$
$$3{l^2} = 2$$
Take Dot with $$\overrightarrow c $$
$$\left( {\overrightarrow a \times \overrightarrow b } \right).\,\overrightarrow c = {\left| {\overrightarrow c } \right|^2} = 2$$
Projection of $$\overrightarrow b $$ or $$\overrightarrow a \times \overrightarrow c = l$$
$${{\left| {\overrightarrow b \,.\,(\overrightarrow a \times \overrightarrow c )} \right|} \over {|\overrightarrow a \times \overrightarrow c |}} = l$$
$$\therefore$$ $$l = {2 \over {\sqrt 6 }} \Rightarrow {l^2} = {4 \over 6}$$
$$3{l^2} = 2$$
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