Let be the angle between a, b where | a|=4, | b|=3, (4,3). · questionr

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Question 192 of 379MediumNUMERICAL

Let

\theta

be the angle between

\vec a,\vec b

where

|\vec a|=4

,

|\vec b|=3

,

\theta\in\left(\dfrac\pi4,\dfrac\pi3\right)

. Then

\big|(\vec a-\vec b)\times(\vec a+\vec b)\big|^2 + 4(\vec a\cdot\vec b)^2

is equal to \_\_ Answer: 576

Solutions · 1

Official solution

\[(\vec a-\vec b)\times(\vec a+\vec b) = \vec a\times\vec b - \vec b\times\vec a = 2(\vec a\times\vec b)\] \[\big|(\vec a-\vec b)\times(\vec a+\vec b)\big|^2 = 4|\vec a\times\vec b|^2 = 4|\vec a|^2|\vec b|^2\sin^2\theta\] \[4(\vec a\cdot\vec b)^2 = 4|\vec a|^2|\vec b|^2\cos^2\theta\] Sum

= 4|\vec a|^2|\vec b|^2(\sin^2\theta+\cos^2\theta) = 4(16)(9) = 576

--- independent of

\theta

.

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