Let vector a=3i-j+2k, vector b=vector a x(i-2k) and vector c=vector b x k.

Question

Let \(\overrightarrow{\mathrm{a}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}-2 \hat{\mathrm{k}})\) and \(\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{\mathrm{k}}.\)

Then the projection of \(\vec{c}-2 \hat{j}\) on \(\vec{a}\) is:

(1) \(3 \sqrt{7}\)

(2) \(\sqrt{14}\)

(3) \(2 \sqrt{14}\)

(4) \(2 \sqrt{7}\)

Answer 1

Correct option is (3) \(2 \sqrt{14}\)   

\(\overrightarrow {\mathrm{b}}=\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}-3 \hat{\mathrm{k}})\)

\(=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 3 & -1 & 2 \\ 1 & 0 & -2\end{array}\right|=2 \hat{i}+8 \hat{j}+\hat{k}\)

\(\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{\mathrm{k}}=8 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}\)

\(\overrightarrow{\mathrm{c}}-2 \hat{\mathrm{j}}=8 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}\)

Projection of \((\hat{\mathrm{i}}-2 \hat{\mathrm{j}})\ \text{on }\overrightarrow{\mathrm{a}}\)

\((\overrightarrow{\mathrm{c}}-2 \hat{\mathrm{j}}) \cdot \hat{\mathrm{a}}=\frac{\langle 8,-4,0\rangle \cdot\langle 3,-1,2\rangle}{\sqrt{14}}\)

\(=\frac{28}{\sqrt{14}}=2 \sqrt{14}\)