What is the scalar projection of {a}={i}-2{j}+{k} on {b}=4{i}-4{j}+7{k}?

Correct Answer is : 19/919/919/9

Correct Answer is : 19/919/919/9

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UPSC NDA 2 2019 Mathematics Solved Paper

Question : 46 of 120

Marks: +1, -0

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

\vec{b}=4\hat{i}-4\hat{j}+7\hat{k}?

Solution:

Concept:

The scalar projection of vector

\vec{a}

onto vector

\vec{b}

is given by

\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Explanation:

Start with

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

and

\vec{b} = 4\hat{i} - 4\hat{j} + 7\hat{k}

Compute the dot product:

\vec{a} \cdot \vec{b} = (1)(4) + (-2)(-4) + (1)(7) = 4 + 8 + 7 = 19

Find the magnitude of

\vec{b}

|\vec{b}| = \sqrt{4^{2} + (-4)^{2} + 7^{2}} = \sqrt{16 + 16 + 49} = \sqrt{81} = 9

Now, the scalar projection is

\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{19}{9}

Answer:

\frac{19}{9}

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