Find the angle between the vectors a = 4, 4, −2 and b = 5, −1, 4.

To find the angle between two vectors, we can use the dot product formula. The dot product of two vectors a and b is given by the equation:

a · b = |a| |b| cos(theta)

where |a| and |b| are the magnitudes of vectors a and b, and theta is the angle between them.

First, let's calculate the magnitudes of vectors a and b:

|a| = sqrt(4^2 + 4^2 + (-2)^2) = sqrt(16 + 16 + 4) = sqrt(36) = 6 |b| = sqrt(5^2 + (-1)^2 + 4^2) = sqrt(25 + 1 + 16) = sqrt(42)

Next, let's calculate the dot product of vectors a and b:

a · b = 4 * 5 + 4 * (-1) + (-2) * 4 = 20 - 4 - 8 = 8

Now, we can substitute the values into the dot product formula to find the angle theta:

8 = 6 * sqrt(42) * cos(theta)

Dividing both sides by 6 * sqrt(42), we get:

cos(theta) = 8 / (6 * sqrt(42))

Now, we can use the inverse cosine function to find the value of theta:

theta = arccos(8 / (6 * sqrt(42)))

Using a calculator, we can find the approximate value of theta to be:

theta ≈ 0.615 radians or 35.26 degrees

Therefore, the angle between vectors a and b is approximately 0.615 radians or 35.26 degrees.