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(A) Let $\vec{a}$ and $\vec{b}$ be the adjacent sides of a parallelogram,where $\vec{a} = 2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\vec{b} = \hat{i}-2 \hat

Vedclass · Accepted Answer

$\frac{3}{7} \hat{i}-\frac{6}{7} \hat{j}+\frac{2}{7} \hat{k}$

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(A) Let $\vec{a}$ and $\vec{b}$ be the adjacent sides of a parallelogram,where $\vec{a} = 2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\vec{b} = \hat{i}-2 \hat{j}-3 \hat{k}$.One diagonal of the parallelogram is given by $\vec{c} = \vec{a} + \vec{b}$.$\vec{c} = (2+1) \hat{i} + (-4-2) \hat{j} + (5-3) \hat{k} = 3 \hat{i} - 6 \hat{j} + 2 \hat{k}$.The magnitude of $\vec{c}$ is $|\vec{c}| = \sqrt{3^2 + (-6)^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7$.The unit vector parallel to the diagonal $\vec{c}$ is $\hat{c} = \frac{\vec{c}}{|\vec{c}|} = \frac{3 \hat{i} - 6 \hat{j} + 2 \hat{k}}{7} = \frac{3}{7} \hat{i} - \frac{6}{7} \hat{j} + \frac{2}{7} \hat{k}$.

Two adjacent sides of a parallelogram are $2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\hat{i}-2 \hat{j}-3 \hat{k}$. Find the unit vector parallel to its diagonal.

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