Find the angle between the vectors u and v if u=<-4,3> and v=<-3,2>, and Round answer to two decimal places.
I found that it is (12*sqrt13)/65, my point of confusion is how do I find the angle?
ever heard of arccos(x) or cos -1x
To find the angle between two vectors, you can use the dot product formula and the magnitude of the vectors.
The dot product of two vectors u and v is defined as the product of their respective components added together. In mathematical notation, the dot product is written as u · v = (u1 * v1) + (u2 * v2).
The magnitude of a vector is the length or size of the vector, which can be calculated using the Pythagorean theorem. In mathematical notation, the magnitude of a vector u is written as ||u|| = sqrt(u1^2 + u2^2).
1. Calculate the dot product of u and v:
u · v = (-4 * -3) + (3 * 2) = 12 + 6 = 18
2. Calculate the magnitudes of u and v:
||u|| = sqrt((-4)^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5
||v|| = sqrt((-3)^2 + 2^2) = sqrt(9 + 4) = sqrt(13)
3. Calculate the angle between u and v using the dot product formula:
cosθ = (u · v) / (||u|| * ||v||)
cosθ = 18 / (5 * sqrt(13))
cosθ ≈ 0.8627
4. Find the angle by taking the inverse cosine (cos^-1) of cosθ:
θ = cos^-1(0.8627)
θ ≈ 30.97 degrees (rounded to two decimal places)
Therefore, the angle between the vectors u and v is approximately 30.97 degrees.