Final answer:
The angle between vectors a = 4i - 3j + k and b = 2i - k is approximately 63 degrees.
Step-by-step explanation:
To find the angle between two vectors, we can use the formula: cos(theta) = (a · b) / (|a||b|), where a · b is the dot product of vectors a and b, and |a| and |b| are the magnitudes of vectors a and b, respectively. In this case, vector a = 4i - 3j + k and vector b = 2i - k. Let's calculate the dot product and magnitudes:
a · b = (4 * 2) + (-3 * 0) + (1 * -1) = 8 - 1 = 7
|a| = sqrt((4^2) + (-3^2) + (1^2)) = sqrt(16 + 9 + 1) = sqrt(26)
|b| = sqrt((2^2) + (-1^2)) = sqrt(4 + 1) = sqrt(5)
Now, let's substitute these values into the formula: cos(theta) = 7 / (sqrt(26) * sqrt(5))
Simplifying further, we have cos(theta) = 7 / (sqrt(130))
To find the angle theta, we can take the inverse cosine of this value: theta = cos^(-1)(7 / sqrt(130))
Using a calculator, we find that the angle theta is approximately 63 degrees.