ind the angle 𝜃 (in degrees) between the vectors. (Round your answer to two decimal places.)

Question

u= 3i + 4j

v= -7i + 5j

𝜃 =

Answer 1

3i +4 = u

-7i +5 = v

dot product of u and v

is the product of their magnitudes times cosine of the angle between them

CosA =dot product divided by the product of their magnitudes

A = Cos^-1(dot product divided by product of magnitudes)

magnitude of U = sqr(3^2 +4^2 = 5

magnitude of v = sqr(49 +1) = sqr50

dot product = 3(-7) +4(5) = -1

CosA = -1 divided by (5sqr50)

A = the angle whose cosine = -1/5sqr50 = 91.62079 approximately

but the angle between 2 vectors is always between 0 and 90, between parallel and perpendicular

so

A = about 90- 1.62 = 88.38 degrees

rounded to 2 decimals

or graph them

u has an angle of 53.13 with the positive x axis

v has an angle of 144.4 with the positive x axis

difference = 144.4 -53.13 = 91.27

= exterior angle. You want the interior angle = 88.73 degrees'

ok slight calculation mistake(s) above but 88.73 is very close to 88.36

Answer 2

u · v = ||u|| ||v|| cos θ