If a =2 i + j +2 k , then the value of | i ×( a × i )|^2+| j ×( a × j )|^2+| k ×( a × k )|^2 is equal to

Correct answer: B. 18 i ×( a × i )=( i · i ) a -( i · a ) i = j +2 k aligned & Similarly, j ×( a × j )=2 i +2 k & k ×( a × k )=2 i + j & \| j +2 k |^2+|^ 2 i +2 k |^2+|^ 2 i + j |^2 & =5+8+5=18 aligned

BITSAT 2024Paper 2

Exam: BITSAT
Year: 2024
Paper: Paper 2
Subject: Mathematics
Chapter: Vector Algebra

BITSAT 2024 · Paper 2 · Mathematics · Vector Algebra

If a =2 i + j +2 k , then the value of | i ×( a × i )|^2+| j ×( a ×...

\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}

, then the value of

|\hat{i} \times(\vec{a} \times \hat{i})|^2+|\hat{j} \times(\vec{a} \times \hat{j})|^2+|\hat{k} \times(\vec{a} \times \hat{k})|^2

is equal to

Options

A $17$
B $18$
C $19$
D $20$

Correct answer

B. $18$

Step-by-step solution

\hat{i} \times(\bar{a} \times \hat{i})=(\hat{i} \cdot \hat{i}) \bar{a}-(\hat{i} \cdot \bar{a}) \hat{i}=\hat{j}+2 \hat{k}

\begin{aligned} & \text{ Similarly, } \hat{j} \times(\bar{a} \times \hat{j})=2 \hat{i}+2 \hat{k} \\ & \hat{k} \times(\bar{a} \times \hat{k})=2 \hat{i}+\hat{j} \\ & \Rightarrow \|\hat{j}+2 \hat{k}|^2+|^{2} \hat{i}+2 \hat{k}|^2+|^{2} \hat{i}+\hat{j}|^2 \\ & =5+8+5=18 \end{aligned}

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