$\mathbf{a}, \mathbf{b}$ are non-collinear vectors, $|\mathbf{a}|=2 \sqrt{2},|\mathbf{b}|=3$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $45^{\circ}$. Then, the lengths of the diagonals of the parallelogram whose adjacent sides are represented by the vectors $5 \mathbf{a}+2 \mathbf{b}$…

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$\mathbf{a}, \mathbf{b}$ are non-collinear vectors, $|\mathbf{a}|=2 \sqrt{2},|\mathbf{b}|=3$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $45^{\circ}$. Then, the lengths of the diagonals of the parallelogram whose adjacent sides are represented by the vectors $5 \mathbf{a}+2 \mathbf{b}$ and $\mathbf{a}-3 \mathbf{b}$ are

Aicharya · Accepted Answer

$\begin{aligned} \mathbf{p} & =5 \mathbf{a}+2 \mathbf{b} \\ \mathbf{q} & =\mathbf{a}-3 \mathbf{b} \\ \mathbf{B D} & =\mathbf{p}+\mathbf{q}=6 \mathbf{a}-\mathbf{b} \\ \mathbf{C A} & =\mathbf{q}-\mathbf{p}=-4 \mathbf{a}-5 \mathbf{b}\end{aligned}$ $\begin{array}{r}\text { Now, }|\mathbf{B D}|=|6 \mathbf{a}-\mathbf{b}| \\ \sqrt{36 \mathbf{a}+|\mathbf{b}|^2-12 \mathbf{a} \mathbf{b}}\end{array}$ $\begin{aligned} & =\sqrt{36|\mathbf{a}|^2+|\mathbf{b}|^2-12|\mathbf{a}||\mathbf{b}| \cos 45^{\circ}} \\ & =\sqrt{36 \times 8+9-12 \times 2 \sqrt{2} \times 3 \times \frac{1}{\sqrt{2}}} \\ & =\sqrt{225}=15\end{aligned}$ and $|\mathbf{C A}|=|-(4 \mathbf{a}+5 \mathbf{b})|$ $=\sqrt{|4 \mathbf{a}|^2+|5 \mathbf{b}|^2+40|\mathbf{a}||\mathbf{b}| \cos 45^{\circ}}$ $\begin{aligned} & =\sqrt{16 \times 8+25 \times 9+40 \times 2 \sqrt{2} \times 3 \times \frac{1}{\sqrt{2}}} \\ & =\sqrt{593}\end{aligned}$ $\therefore$ Lengths of the diagonals are 15 and $\sqrt{593}$.

$\mathbf{a}, \mathbf{b}$ are non-collinear vectors, $|\mathbf{a}|=2 \sqrt{2},|\mathbf{b}|=3$ and the angle…

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