Mathematics - Vector Question with Solution | TestHub

Given that $\vec{a}\times (2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times \vec{b}$ $\Rightarrow (\vec{a}+\vec{b})\times (2\hat{i}+3\hat{j}+4\hat{k})=\overrightarrow{0}$ But $\vec{a}+\vec{b}\ne 0$ and $2\hat{i}+3\hat{j}+4\hat{k}\ne 0$ $\therefore (\vec{a}+\vec{b})\Vert (2\hat{i}+3\hat{j}+4\hat{k})$. Let $\vec{a}+\vec{b}=\lambda (2\hat{i}+3\hat{j}+4\hat{k})$ Also given that $|\vec{a}+\vec{b}|=\sqrt{29}\Rightarrow \lambda =\pm 1$ $\therefore \vec{a}+\vec{b}=\pm (2\hat{i}+3\hat{j}+4\hat{k})$ So, $(\vec{a}+\vec{b})\cdot (-7\hat{i}+2\hat{i}+3\hat{k})=\pm 4$