NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

If $\mathbf{a} = 4\hat{i} + 6\hat{j}$ and $\mathbf{b} = 3\hat{i} + 4\hat{k}$ , then the vector form of the component of $\mathbf{a}$ along $\mathbf{b}$ is:

Choose the correct answer:

Explanation

1. Formula:

The vector component (projection) of $\mathbf{a}$ along $\mathbf{b}$ is given by:

\text{Vector Component} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \right) \mathbf{b}

2. Finding Dot Product ( $\mathbf{a} \cdot \mathbf{b}$ ):

\mathbf{a} = 4\hat{i} + 6\hat{j} + 0\hat{k}

\mathbf{b} = 3\hat{i} + 0\hat{j} + 4\hat{k}

\mathbf{a} \cdot \mathbf{b} = (4 \times 3) + (6 \times 0) + (0 \times 4)

\mathbf{a} \cdot \mathbf{b} = 12 + 0 + 0 = 12

3. Finding Magnitude Squared ( $|\mathbf{b}|^2$ ):

|\mathbf{b}| = \sqrt{3^2 + 0^2 + 4^2}

|\mathbf{b}| = \sqrt{9 + 16} = \sqrt{25} = 5

|\mathbf{b}|^2 = 25

4. Final Solving:

\text{Vector Component} = \left( \frac{12}{25} \right) \mathbf{b}

\text{Vector Component} = \frac{12}{25} (3\hat{i} + 4\hat{k})

Conclusion:

Correct option is (d).