Question

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = 2, 0, 1 , b = 4, −2, 0.

Exact =
Approximate degree =

Answer 1

To find the angle between vectors a and b, we can use the dot product formula. After calculating the dot product and finding the magnitudes of the vectors, we can solve for the angle using inverse cosine. The exact expression for the angle is arccos((4sqrt(5))/5), which is approximately 51 degrees.

Step-by-step explanation:

To find the angle between two vectors, we can use the dot product formula: A · B = |A| |B| cos(theta).

First, we can calculate the dot product of the given vectors a and b: (2)(4) + (0)(-2) + (1)(0) = 8.

Next, we can find the magnitudes of both vectors: |a| = sqrt(2^2 + 0^2 + 1^2) = sqrt(5) and |b| = sqrt(4^2 + (-2)^2 + 0^2) = sqrt(20).

Using these values, we can solve for the angle theta: 8 = sqrt(5)sqrt(20)cos(theta). Solving for cos(theta), we get cos(theta) = 8/(sqrt(5)sqrt(20)) = 8/(2sqrt(5)) = 4/sqrt(5) = (4sqrt(5))/5.

Finally, we can find the exact expression for the angle: theta = arccos((4sqrt(5))/5).

To approximate the angle to the nearest degree, we can use a calculator and find that the angle is approximately 51 degrees.