Let vector a = 2i + j - k, vector b = ((vector a x (i + j)) x i) x i. Then the square of the projection

Question

Let \(\vec{a}=2 \hat{i}+\hat{j}-\hat{k}, \vec{b}=((\vec{a} \times(\hat{i}+\hat{j})) \times \hat{i}) \times \hat{i}.\) Then the square of the projection of \( \vec{a} \,on\,\vec{b}\) is :

(1) \(\frac{1}{5}\)

(2) 2

(3) \(\frac{1}{3}\)

(4) \(\frac{2}{3}\)

Answer 1

Correct option is (2) 2

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

\(=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\)

\((\vec{a} \times(\hat{i} \times \hat{j})) \times \hat{i}=\hat{k}+\hat{j}\)

\(((\vec{a} \times(\hat{i} \times \hat{j})) \times i) \times \hat{i}=\hat{j}-\hat{k}\)

projection of \( \vec{a}\, on \,\hat{b}=\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\)

\(=\frac{1+1}{\sqrt{2}}=\sqrt{2}\)