find the angle between u(-1,4) and V(3,-4)
To find the angle between two vectors, you can use the dot product formula:
cos(theta) = (u · v) / (||u|| * ||v||)
Where:
- u · v is the dot product of vectors u and v
- ||u|| and ||v|| are the magnitudes (or lengths) of vectors u and v, respectively
Let's solve the problem step-by-step.
Step 1: Calculate the dot product (u · v)
The dot product of two vectors u = (-1, 4) and v = (3, -4) is:
u · v = (-1 * 3) + (4 * -4)
= -3 - 16
= -19
Step 2: Calculate the magnitudes ||u|| and ||v||
The magnitude of a vector (x, y) can be calculated using the formula:
||v|| = sqrt(x^2 + y^2)
For vector u = (-1, 4):
||u|| = sqrt((-1)^2 + 4^2)
= sqrt(1 + 16)
= sqrt(17)
For vector v = (3, -4):
||v|| = sqrt(3^2 + (-4)^2)
= sqrt(9 + 16)
= sqrt(25)
= 5
Step 3: Calculate the angle theta using the formula:
cos(theta) = (u · v) / (||u|| * ||v||)
theta = arc cos((u · v) / (||u|| * ||v||))
= arc cos(-19 / (sqrt(17) * 5))
Using a calculator or software that can compute arccosine, we find:
theta ≈ 135.93 degrees
Therefore, the angle between vectors u(-1, 4) and v(3, -4) is approximately 135.93 degrees.