If $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ are sides of a parallelogram, then a unit vector is parallel to one of the diagonals of the parallelogram is

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Aicharya · Accepted Answer

Let the position vector $ \mathbf{O A}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \mathbf{O B}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}} $

$\therefore$ Diagonal vector $ \begin{aligned} & \mathrm{r} \mathbf{O C}=\mathbf{O A}+\mathbf{A C} \\ & =\mathbf{O A}+\mathbf{O B} \quad[\because \mathbf{O B} \| \mathbf{A C}] \\ & =\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}+3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}} \\ & =4(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \end{aligned} $ $\therefore$ Unit vector of a diagonal $ =\frac{4(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})}{\sqrt{4^2+4^2+4^2}}=\frac{(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})}{\sqrt{3}} $

If $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 3 \hat{\mathbf{i}}+2…

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