How do you find the angle, θ, between the vectors? If needed, round your answer to the nearest tenth.


1.) u = 2i - 3j
v = i - 2j

2.) u = <3, 2>
v = <4, 0>

1.

dot product
u ∙ v = |u| |v| cosØ, where Ø is the angle between them, so
<2,-3> ∙ <1,-2> = √13 √5 cosØ
2+6 = √13√5cosØ
cosØ = 8/(√13√5) = .99227...
Ø = appr 7.1°

do #2 the same way

To find the angle θ between two vectors, you can use the dot product formula and the magnitude of the vectors.

1.) Given u = 2i - 3j and v = i - 2j, we can calculate the dot product and magnitudes.

a. Dot Product:
The dot product of two vectors u and v is given by the formula: u · v = |u| |v| cos(θ), where |u| and |v| are the magnitudes of the vectors.

|u| = sqrt(2^2 + (-3)^2) = sqrt(4 + 9) = sqrt(13)
|v| = sqrt(1^2 + (-2)^2) = sqrt(1 + 4) = sqrt(5)

To find u · v, we multiply the respective elements and sum them up:
u · v = 2(1) + (-3)(-2) = 2 + 6 = 8

b. Magnitude:
The magnitude of a vector v is given by the formula: |v| = sqrt(vx^2 + vy^2), where vx and vy are the components of the vector.

Using the magnitudes found earlier, |u| = sqrt(13) and |v| = sqrt(5).

c. Angle:
Using the dot product formula mentioned earlier, we can rearrange the formula to solve for θ:
cos(θ) = (u · v) / (|u| |v|)

Plugging in the values we found:
cos(θ) = 8 / (sqrt(13) * sqrt(5))

Finally, we can solve for θ:
θ = arccos(8 / (sqrt(13) * sqrt(5)))

Just substitute the values in a scientific calculator and round the answer to the nearest tenth.

2.) Given u = <3, 2> and v = <4, 0>, we can apply the same steps as in the previous example.

a. Dot Product:
u · v = (3)(4) + (2)(0) = 12 + 0 = 12

b. Magnitude:
|u| = sqrt(3^2 + 2^2) = sqrt(9 + 4) = sqrt(13)
|v| = sqrt(4^2 + 0^2) = sqrt(16 + 0) = sqrt(16) = 4

c. Angle:
Using the dot product formula mentioned earlier, we can rearrange the formula to solve for θ:
cos(θ) = (u · v) / (|u| |v|)

Plugging in the values we found:
cos(θ) = 12 / (sqrt(13) * 4)

Finally, we can solve for θ:
θ = arccos(12 / (sqrt(13) * 4))

Just substitute the values in a scientific calculator and round the answer to the nearest tenth.