Find the Degree Measure of the angle between the vectors.
U=-4i+4j
V=-4i+4j
your two vectors are the same, so the angle between them is 0
check your typing.
Haha, right, thanks.
To find the degree measure of the angle between two vectors, U and V, you can use the dot product formula and inverse cosine (arccos) function. Here's how:
1. Start by calculating the dot product of U and V. The dot product (U · V) of two vectors U = (Ux, Uy) and V = (Vx, Vy) is given by:
U · V = Ux * Vx + Uy * Vy
In this case:
U = -4i + 4j = (-4, 4)
V = -4i + 4j = (-4, 4)
So, calculate the dot product:
U · V = -4 * -4 + 4 * 4 = 16 + 16 = 32
2. Next, calculate the magnitudes (or lengths) of the vectors U and V. The magnitude (or length) of a vector (x, y) is given by:
|U| = √(Ux^2 + Uy^2)
For U = (-4, 4):
|U| = √((-4)^2 + 4^2) = √(16 + 16) = √32 = 4√2
For V = (-4, 4):
|V| = √((-4)^2 + 4^2) = √(16 + 16) = √32 = 4√2
3. Now, substitute the previously calculated values into the formula for the angle between two vectors:
cos θ = (U · V) / (|U| * |V|)
Using the dot product and magnitudes calculated earlier:
cos θ = 32 / (4√2 * 4√2) = 32 / (4 * 2 * 4) = 32 / 32 = 1
4. Finally, find the inverse cosine (arccos) of the result to get the angle measure in radians:
θ = arccos(1) = 0 radians
To convert from radians to degrees, multiply by 180/π (approximately 57.2958):
θ = 0 * (180/π) = 0 degrees
Therefore, the degree measure of the angle between the vectors U = -4i + 4j and V = -4i + 4j is 0 degrees.