In each of the following, find the angle formed by u and v:
(a) u = [2, 1] and v = [1,−3]
(b) u = [−1, 0, 1] and v = [0, 2,−2]
I will do the first, you do the second
u•v = |u||v|cosØ, where Ø is the angle between them
cos Ø = (2 - 3)/(√5√10) = -1/√50 = -..14142
Ø = 81.87°
let me know what you get for the other one.
If you want your angle in radians, make sure your calculator is set to RAD
thanks
To find the angle formed by two vectors, we will use the dot product formula and the magnitude formula.
(a) u = [2, 1] and v = [1,−3]:
First, calculate the dot product of u and v.
The dot product of two vectors, u and v, is calculated as:
u · v = u1 * v1 + u2 * v2
In this case, u1 = 2, u2 = 1, v1 = 1, and v2 = -3.
So, u · v = (2 * 1) + (1 * -3) = 2 - 3 = -1.
Next, calculate the magnitude of vectors u and v.
The magnitude of a vector is calculated as:
|u| = √(u1^2 + u2^2)
For vector u, |u| = √(2^2 + 1^2) = √(4 + 1) = √5.
For vector v, |v| = √(1^2 + (-3)^2) = √(1 + 9) = √10.
Finally, to find the angle between vectors u and v, we will use the dot product and magnitude:
cosθ = (u · v) / (|u| * |v|)
Therefore, θ = arccos((u · v) / (|u| * |v|))
In this case, the angle θ = arccos(-1 / (√5 * √10)).
(b) u = [−1, 0, 1] and v = [0, 2,−2]:
First, calculate the dot product of u and v.
The dot product of two vectors, u and v, is calculated as:
u · v = u1 * v1 + u2 * v2 + u3 * v3
In this case, u1 = -1, u2 = 0, u3 = 1, v1 = 0, v2 = 2, and v3 = -2.
So, u · v = (-1 * 0) + (0 * 2) + (1 * -2) = 0 + 0 - 2 = -2.
Next, calculate the magnitude of vectors u and v.
The magnitude of a vector is calculated as:
|u| = √(u1^2 + u2^2 + u3^2)
For vector u, |u| = √((-1)^2 + 0^2 + 1^2) = √(1 + 1) = √2.
For vector v, |v| = √(0^2 + 2^2 + (-2)^2) = √(0 + 4 + 4) = √8 = 2√2.
Finally, to find the angle between vectors u and v, we will use the dot product and magnitude:
cosθ = (u · v) / (|u| * |v|)
Therefore, θ = arccos((u · v) / (|u| * |v|))
In this case, the angle θ = arccos(-2 / (2√2)).