# There are two vectors a and b.If a=(1,2,3) and b=(4,5,6), what is: a) A vector that is perpindicular to both vectors, and has a magnitude of 4? b) What is the angle between the two vectors?

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### 1 Answer

A vector that is perpendicular to both is the cross product of a and b

`a xx b = <2*6-3*5,3*4-1*6,1*5-2*4> = <-3,6,-3>`

So any vector in this direction will be perpendicular to both a and b, and now we just need to scale it to the right length.

`|<-3,6,-3>|=sqrt((-3)^2+6^2+(-3)^2)=3sqrt(6)`

Thus we want to multiply our vector `<-3,6,-3>` by `(4)/(3sqrt(6))`

So a vector perpendicular to a and b, and with length 4 is:

`<-(2)/(3) sqrt(6), (4)/(3) sqrt(6), -(2)/(3) sqrt(6)>`

(and the other vector that works is the same length, opposite direction: `<(2)/(3) sqrt(6), -(4)/(3) sqrt(6), (2)/(3) sqrt(6)>` )

The angle between the two vectors can be gotten using the formula:

`a*b=|a| |b| "cos" theta`

`a*b = 1*4+2*5+3*6=32`

`|a|=sqrt(1^2+2^2+3^2)=sqrt(14)`

`|b|=sqrt(4^2+5^2+6^2)=sqrt(77)`

So:

`32 = 7 sqrt(22) "cos" theta`

`(32)/(7 sqrt(22)) = "cos" theta`

`theta = .2257` (in radians)