If a and b are vectors such that | a + b |= 29 and a ×(2 i +3 j +4 k )=(2 i +3 j +4 k ) × b , then a possible value of ( a + b ) ·(-7 i +2 j +3 k ) is

Correct answer: C. 4 Given that a ×(2 i +3 j +4 k )=(2 i +3 j +4 k ) × b ( a + b ) ×(2 i +3 j +4 k )= 0 But a + b ≠ 0 and 2 i +3 j +4 k ≠ 0 ( a + b ) \|(2 i +3 j +4 k ) . Let a + b =λ(2 i +3 j +4 k ) Also given that | a + b |= 29 λ=± 1 a + b =±(2 i +3 j +4 k ) So, ( a + b ) ·(-7 i +2 i +3 k )=± 4

JEE Advanced 2012Paper 2

Exam: JEE Advanced
Year: 2012
Paper: Paper 2
Subject: Mathematics
Chapter: Vector Algebra

JEE Advanced 2012 · Paper 2 · Mathematics · Vector Algebra

If a and b are vectors such that | a + b |= 29 and a ×(2 i +3 j +4...

\vec{a}

and

\vec{b}

are vectors such that

|\vec{a}+\vec{b}|=\sqrt{29}

and

\vec{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{b}

, then a possible value of

(\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})

Options

Correct answer

C. 4

Step-by-step solution

Given that

\vec{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{b}

\Rightarrow(\vec{a}+\vec{b}) \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=\overrightarrow{0}

But

\vec{a}+\vec{b} \neq 0

and

2 \hat{i}+3 \hat{j}+4 \hat{k} \neq 0

\therefore(\vec{a}+\vec{b}) \|(2 \hat{i}+3 \hat{j}+4 \hat{k})

. Let

\vec{a}+\vec{b}=\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})

Also given that

|\vec{a}+\vec{b}|=\sqrt{29} \Rightarrow \lambda=\pm 1

\therefore \vec{a}+\vec{b}=\pm(2 \hat{i}+3 \hat{j}+4 \hat{k})

So,

(\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{i}+3 \hat{k})=\pm 4

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