If the angle between (a,2) and (a,-2) has measure π/3, Find a.
Answers
Answer:
you mean, angle between \bar{a}
a
ˉ
and \bar{b}
b
ˉ
is π/3 then we have to find angle between 2\bar{a}2
a
ˉ
and -3\bar{b}−3
b
ˉ
use dot product concepts ,
we know, if A and B are two vectors and angle between them is \thetaθ
then, A.B=|A||B|cos\thetaA.B=∣A∣∣B∣cosθ
or, \theta=cos^{-1}\left(\frac{A.B}{|A||B|}\right)θ=cos
−1
(
∣A∣∣B∣
A.B
)
case 1 :- \frac{\pi}{3}=cos^{-1}\frac{\bar{a}.\bar{b}}{|a||b|}
3
π
=cos
−1
∣a∣∣b∣
a
ˉ
.
b
ˉ
........(1)
case 2 : Let angle between 2\bar{a}2
a
ˉ
and -3\bar{b}−3
b
ˉ
is \thetaθ
then, \theta=cos^{-1}\frac{2\bar{a}.-3\bar{b}}{|2a||-3b|}θ=cos
−1
∣2a∣∣−3b∣
2
a
ˉ
.−3
b
ˉ
=cos^{-1}\frac{-6\bar{a}.\bar{b}}{6|a||b|}cos
−1
6∣a∣∣b∣
−6
a
ˉ
.
b
ˉ
=\pi-cos^{-1}(1)\frac{\bar{a}.\bar{b}}{|a||b|}π−cos
−1
(1)
∣a∣∣b∣
a
ˉ
.
b
ˉ
from equation (1),
= π - π/3 = 2π/3
hence, answer is 2π/3