u=(2 cos pi/4)i + (2 sin pi/4)j, v = (cos 3pi/2)i + (sin 3pi/2)j
find the angle theta between the two
I'm at a lost on what to do.
To find the angle theta between two vectors, you can use the dot product formula:
θ = cos^(-1)((u·v) / (|u| |v|))
where u and v are the given vectors, · denotes the dot product, |u| represents the magnitude (or length) of vector u, and |v| represents the magnitude of vector v.
Let's calculate it step by step.
Given:
u = (2 cos π/4)i + (2 sin π/4)j
v = (cos (3π/2))i + (sin (3π/2))j
First, we need to find the dot product (u·v):
u·v = (2 cos π/4)(cos (3π/2)) + (2 sin π/4)(sin (3π/2))
To simplify further, recall these trigonometric identities:
cos (π/4) = √2/2
sin (π/4) = √2/2
Using these identities, we can rewrite the dot product as:
u·v = (2 √2/2)(0) + (2 √2/2)(-1)
Simplifying further, we have:
u·v = 0 - √2 = -√2
Next, we need to find the magnitudes of the vectors (|u| and |v|):
|u| = √((2 cos π/4)^2 + (2 sin π/4)^2) = √(2^2 + 2^2) = √8 = 2√2
|v| = √((cos (3π/2))^2 + (sin (3π/2))^2) = √((-1)^2 + 0^2) = √1 = 1
Now, substitute the values into the formula to find the angle theta:
θ = cos^(-1)((u·v) / (|u| |v|))
θ = cos^(-1)(-√2 / (2√2)(1))
θ = cos^(-1)(-1/√2)
Finally, use a calculator to find the inverse cosine of -1/√2:
θ ≈ 135 degrees
Therefore, the angle theta between the given vectors u and v is approximately 135 degrees.