Question

Find the angle between the vectors a = (2, 3) and b = (3, -1). First, find the exact expression and then approximate it to the nearest degree.

Answer 1

To find the angle between two vectors, a = (2, 3) and b = (3, -1), we can use the formula angle = arccos((a・b) / (|a||b|)). The angle can be calculated step by step by finding the dot product of the vectors, calculating the magnitudes of the vectors, and then plugging the values into the formula. The angle between vectors a and b is approximately 55.48 degrees.

Step-by-step explanation:

To find the angle between two vectors, a = (2, 3) and b = (3, -1), we can use the formula:

angle = arccos((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.

Let's calculate it step by step:

1. Calculate the dot product of vectors a and b: a・b = 2*3 + 3*(-1) = 6 - 3 = 3
2. Calculate the magnitudes of vectors a and b: |a| = sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13) and |b| = sqrt(3^2 + (-1)^2) = sqrt(9 + 1) = sqrt(10)
3. Plug the values into the formula: angle = arccos(3 / (sqrt(13) * sqrt(10)))
4. Calculate the angle using a calculator: angle ≈ 55.48 degrees

The angle between vectors a and b is approximately 55.48 degrees.