Let a =2 i + j - k , b =(( a ×( i + j )) × i ) × i . Then the square of the projection of a on b is :

Correct answer: C. 2 aligned & a ×( i + j )=| array ccc i & j & k 2 & 1 & -1 1 & 1 & 0 array | & = i - j + k & ( a ×( i × j )) × i = k + j & (( a ×( i × j )) × i) × i = j - k & projection of a on b = a · b | b | & = 1+1 2 = 2 aligned

JEE Main 20246 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 6 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2024 · 6 April · Evening Shift · Mathematics · Vector Algebra

Let a =2 i + j - k , b =(( a ×( i + j )) × i ) × i . Then the...

Let

\overrightarrow{\mathrm{a}}=2 \hat{i}+\hat{j}-\hat{k}, \overrightarrow{\mathrm{b}}=((\overrightarrow{\mathrm{a}} \times(\hat{i}+\hat{j})) \times \hat{i}) \times \hat{i}

. Then the square of the projection of

\overrightarrow{\mathrm{a}}

\overrightarrow{\mathrm{b}}

is :

Options

A $\frac{1}{3}$
B $\frac{2}{3}$
C2
D $\frac{1}{5}$

Correct answer

C. 2

Step-by-step solution

\begin{aligned} & \vec{a} \times(\hat{i}+\hat{j})=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -1 \\ 1 & 1 & 0\end{array}\right| \\ & =\hat{i}-\hat{j}+\hat{k} \\ & (\vec{a} \times(\hat{i} \times \hat{j})) \times \hat{i}=\hat{k}+\hat{j} \\ & ((\vec{a} \times(\hat{i} \times \hat{j})) \times i) \times \hat{i}=\hat{j}-\hat{k} \\ & \operatorname{projection} \text{\text{ of } } \vec{a} \text{ on } \hat{b}=\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \\ & =\frac{1+1}{\sqrt{2}}=\sqrt{2}\end{aligned}

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