Physics - Vectors Question with Solution | TestHub

Two vectors must be perpendicular if their dot product is zero. Let, $\begin{array}{rl}\overrightarrow{\mathbf{a}}& =2\hat{\mathbf{i}}+3\hat{\mathbf{j}}+8\hat{\mathbf{k}}\\ \overrightarrow{\mathbf{b}}& =4\hat{\mathbf{j}}-4\hat{\mathbf{i}}+\alpha \hat{\mathbf{k}}\\ & =-4\hat{\mathbf{i}}+4\hat{\mathbf{j}}+\alpha \hat{\mathbf{k}}\end{array}$ According to the above hypothesis $\overrightarrow{\mathbf{a}}\perp \overrightarrow{\mathbf{b}}$ $\Rightarrow \overrightarrow{\mathbf{a}}\cdot \overrightarrow{\mathbf{b}}=0$ $\begin{array}{lc}\Rightarrow & (2\hat{\mathbf{i}}+3\hat{\mathbf{j}}+8\hat{\mathbf{k}})\cdot (-4\hat{\mathbf{i}}+4\hat{\mathbf{j}}+\alpha \hat{\mathbf{k}})=0\\ \Rightarrow & -8+12+8\alpha =0\\ \Rightarrow & 8\alpha =-4\\ \therefore & \alpha =-\frac{4}{8}=-\frac{1}{2}\end{array}$ NOTE: $\overrightarrow{\mathbf{a}}\cdot \overrightarrow{\mathbf{b}}=ab\cos \theta$. Here, $a$ and $b$ are always positive as they are the magnitudes $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$.