Let \(\mathbf{u}=-2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and…
Let \(\mathbf{u}=-2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{v}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\). Then the component of \(\mathbf{v}\) on \(\mathbf{u}\) is
\(\frac{4}{3}\)
\(\frac{-4}{3}\)
\(\frac{-2}{3}\)
\(\frac{2}{3}\)
Solution
Given vectors are \(\mathbf{u}=-2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{v}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\)
\(\therefore\) Component of \(\mathbf{v}\) on \(\mathbf{u}\) is \(=\frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{u}|}=\frac{-2-4+2}{\sqrt{4+4+1}}=\frac{-4}{3}\)
Hence, option (b) is correct.