The length of longer diagonal of the parallelogram constructed on. 5 a + 2 b 5a+2b and a − 3 b a-3b, if it is given that | a | = 2 √ 2 , | b | = 3 |a|=22,|b|=3 and the angle between a a and b b is π 4 π4 is

The length of longer diagonal of the 

parallelogram constructed on. $5a+2b$ and

$a-3b$, if it is given that $\left|a\right|=2\sqrt{2},\left|b\right|=3$ and the

angle between $a$ and $b$ is $\frac{\pi }{4}$ is 

A

$15$

B

$\sqrt{113}$

C

$\sqrt{593}$

D

$\sqrt{$ \sqrt{369}$}$

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SOLUTION

The Correct Answer is $\Rightarrow$ $3$

Given that,$\left|a\right|=2\sqrt{2},\left|b\right|=3$

The longer vector is $5a+2b+a-3b=6a-b$

Length of one diagonal

$=|6a-b|$

$=\sqrt{36a^2+b^2-2\times 6\left|a\right|\cdot \left|b\right|\cdot {\cos 45}^0}$

$=\sqrt{36\times 8+9-12\times 2\sqrt{2}\times 3\times \frac{1}{\sqrt{2}}}$

$=\sqrt{288+9-12\times 6}=\sqrt{225}=15$

Other diagonal is $4a+5b$.

Its length is

$=\sqrt{{\left(4a\right)}^2+{\left(5b\right)}^2+2x\left|4a\right|\left|5b\right|{\cos 45}^0}$

$=\sqrt{16\times 8+25\times 9+40\times 6}=\sqrt{593}$