Use an algebraic method to find the angle between the vectors. Use a calculator to approximate exact answers when appropriate. u=<−15,−8> and v=<-1,5> ​(Round to three decimal places as​ needed.)

Question

Use an algebraic method to find the angle between the vectors. Use a calculator to approximate exact answers when appropriate. u=<−15,−8> and v=<-1,5> ​(Round to three decimal places as​ needed.)

Answer 1

u=<−15,−8> and v=<-1,5>

The simplest way is to use a dot product. If θ is the angle between u and v, then

cos θ = (u.v)/(||u||*||v||)

In this case:

u.v = (-15)(-1) + (-8)(5) = -25

||u|| = sqrt((-15)^2 + (-1)^2) = sqrt(289)

||v|| = sqrt((-8)^2 + 5^2) = sqrt(26)

...giving cos θ ≈ -0.2884. Therefore θ ≈ 1.8634 radians or 106.7626 degrees.

Answer 2

<-15,-8> <-1,5>

angle = tan^-1(5/1) + tan^-1(8/15) using reference angles

= 106.6731622 degrees

= about 106.673 degrees

= about .5926 pi radians

= about .593pi radians

= about 1.861798 radians

= about 1.862 radians

it helps to draw a rough sketch of

the angle between the 2 vectors

then it appears a little more than 90 degrees

and a little less than 2 radians

Answer 3

<-15, -8> creates angle α with x-axis

<-1, 5> cxreates angle β with x-axis

tan α = 5, tan β = 8 / 15

tan (α + β) = (tan α + tan β) / (1 - tan α tan β)

tan (α + β) = (5 + (8/15)) / (1 - (5)(8/15))

tan (α + β) = (83/15) / (8/3)

tan (α + β) = 2.075

α + β = 1.1217108 rad