vectors u = <5, 8> & v=<20,5> , find angle between u & v and projection where proj(v)u=(u*v/|v|^2)v thank you for help!

1 Answer | Add Yours

academicsfirst's profile pic

academicsfirst | High School Teacher | (Level 2) Adjunct Educator

Posted on

We are asked to find the angle between vectors u and v and the projection where proj(v)u = (u*v/|v| ^ 2)v.

u= <5,8> and v = <20,5>.

We will first find the angle between vectors u and v.

The formula is cos(theta) = (u *v)/(|u| *|v|)

=> cos (theta) =

(5*20) + (8 * 5)/ [(sqrt 5^2 + 8 ^2) * (sqrt 20 ^ 2 + 5 ^ 2)]

=> cos (theta) =

(100 + 40)/(sqrt 25 +64) *(sqrt 400 + 25)

=> cos (theta) = (140) /(sqrt 89) *(sqrt 425)

=> cos (theta) = (140) / (sqrt 37825)

=> cos (theta) = (140)/(194.4865)

=>  The angle between the two vectors is approximately 43.96 degrees.

To find the projection we use the following formula which we  were given:

=> projection  (v)u =  (u * v/ |v| ^2) v

=> projection (v)u =

{[(5 * 20) + (8 * 5)/[sqrt  (20 ^2 + 5 ^2)] }* (20, 5)

=> projection (v)u = [(140)/(sqrt 425) ] * (20,5)

=> projection(v)u  = 6.79 * ( 20, 5)

=> projection(v)u = ( 135.8, 33.95 )

The angle between the two vectors is approximately 43.96 degrees.

The projection(v)u = ( 135.8, 33.95 ).

We’ve answered 318,911 questions. We can answer yours, too.

Ask a question