# Relative to an origin O, th position vectors of the points A, B and C are given by OA (2,3,-6), OB (0,-6,8), OC (-2,5,-2) (i) Find angle AOB (ii) Fing nthe vector which is in the same direction...

Relative to an origin O, th position vectors of the points A, B and C are given by

OA (2,3,-6), OB (0,-6,8), OC (-2,5,-2)

(i) Find angle AOB

(ii) Fing nthe vector which is in the same direction as AC and has magnitude 30.

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### 1 Answer

OA (2,3,-6), OB (0,-6,8), OC (-2,5,-2)

vector OA=**a**=2i+3j-6k

vector OB=**b**=0i-6b+8k

`a*b=|a||b|cos(theta)` ,where `theta` is angle BOA.

`0-16-48=sqrt(4+9+36)sqrt(36+64)cos(theta)`

`cos(theta)=(-64)/{sqrt(49)sqrt(100)}`

`cos(theta)=(-64)/70`

`cos(theta)=-.9142`

`theta=180-23.9`

`=156.1^o`

`` vector AC=(-2-2)i+(5-3)j+(-2+6)k

=-4i+2j+4k

`|AC|=sqrt((-4)^2+2^2+4^2)`

`=sqrt(34)`

let ED is vector parallel to AC and `|ED|=30`

`ED=lambda AC`

`|ED|=|lambdaAC|`

`+-30=lambdasqrt(34)`

`lambda=(+-30)/sqrt(34)`

`ED={(+-30)/sqrt(34)}{-4i+2j+4k}`