Let u = 6i + 8j and v = 4i − j. (a) Find the angle between u and v. (Round your answer to one decimal place.) ° (b) Find the component of u along v. (c) Find projvu.

Question

Let u = 6i + 8j

 and v = 4i − j.

(a) Find the angle between u and v. (Round your answer to one decimal place.)

 °

(b) Find the component of u along v.

 

(c) Find proj_vu.

Answer 1

Hi,

Let u = 6i + 8j ------> (6,8)

 and v = 4i − j.------->(4,-1)

First, graph u and v on the coordinate plane then connect the end of u and v to each other and named w

Second : || u || = √6^2 + 8^2 = 10 AND ||v|| = √4^2 + (-1)^2 = √17 AND ||w|| = √(6-4)^2 + (8+1)^2 = √85

Third: use cosine law-------> cosγ = [( 10 )^2 + (√17)^2 - (√85)^2] / 2 *10 * √17) = 0.39

Fourth: use calculator when cosγγ = 0.39 ----> γ = 67.17 degree

b) Find the component of u along first method: v = | u | cosγ = 10 *0.39 = 3.9

Second method: component of u along v:

1) find u.v ( dot product) = 6*4 + 8 * (-1) = 24 -8 = 16 2) |v| = √17 3) 16/√17=3.88057 ≈3.9

c) Find proj_vu = [u.v/||v||² ] . v

first, find [u.v]/ ||v||² = [ 6*4 -8*1] / [||v|| = √4^2 + (-1)^2 = √17]² = 16/ 17 =

Now proj_vu = 16/ 17<4,-1> = < 64/17 , -16/17 >

I hope it is useful,

Minoo