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UPSC NDA I 2023 Mathematics Solved Paper

Question Numbers: 67-68

Consider the following for the next two (02) items that follow :

Let a vector

\vec{a}=4\hat{i}-8\hat{j}+\hat{k}

make angles α, β, γ with the positive directions of x, y, z axes respectively.

Question : 67 of 120

Marks: +1, -0

What is cosα equal to ?

Solution:

Concept:

Direction cosine cosα is the ratio of the x-component of a vector to its magnitude.

Explanation:

Given vector

\vec{a} = 4\hat{i} - 8\hat{j} + \hat{k}

Its magnitude is

|\vec{a}| = \sqrt{4^2 + (-8)^2 + 1^2} = \sqrt{16 + 64 + 1} = \sqrt{81} = 9

Therefore,

\cos \alpha = \frac{4}{|\vec{a}|} = \frac{4}{9}

Answer:

\frac{4}{9}

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