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UPSC NDA I 2017 Mathematics Solved Paper

Question : 56 of 120

Marks: +1, -0

\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}

\vec{b}=3\hat{i}+2\hat{j}-λ\hat{k}

perpendicular, then what is the value of λ

Solution:

Concept:

Two vectors are perpendicular if and only if their dot product is zero.

Explanation:

Given vectors are

\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}

and

\vec{b}=3\hat{i}+2\hat{j}-\lambda\hat{k}

They are perpendicular, so

\vec{a}\cdot\vec{b}=0

Compute the dot product:

(2\hat{i}+3\hat{j}+4\hat{k})\cdot(3\hat{i}+2\hat{j}-\lambda\hat{k})

2\cdot3 + 3\cdot2 + 4\cdot(-\lambda)

6 + 6 - 4\lambda

12 - 4\lambda

Set this equal to zero:

12 - 4\lambda = 0

Solve for

\lambda

4\lambda = 12

, hence

\lambda = 3

Answer:

\lambda = 3

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