Question

If the angle between (a,2) and (a,-2) has measure π/3, Find a.

Answer 1

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