Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈2, 1〉, 〈-3, 1〉

Verified step by step guidance

1. Identify the two vectors given: \( \mathbf{u} = \langle 2, 1 \rangle \) and \( \mathbf{v} = \langle -3, 1 \rangle \).
2. Recall the formula for the angle \( \theta \) between two vectors \( \mathbf{u} \) and \( \mathbf{v} \): \[\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}\]
3. Calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \) using the formula: \[\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2\] where \( u_1, u_2 \) and \( v_1, v_2 \) are the components of \( \mathbf{u} \) and \( \mathbf{v} \) respectively.
4. Find the magnitudes of each vector using: \[\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}\] and \[\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}\]
5. Substitute the dot product and magnitudes into the cosine formula, then use the inverse cosine function to find the angle \( \theta = \cos^{-1} \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \right) \). Finally, round your answer to two decimal places.

Key Concepts

Dot Product of Vectors

The dot product is a scalar value obtained by multiplying corresponding components of two vectors and summing the results. For vectors 〈a, b〉 and 〈c, d〉, the dot product is ac + bd. It is essential for finding the angle between vectors.

Magnitude of a Vector

The magnitude (or length) of a vector 〈x, y〉 is calculated using the formula √(x² + y²). It represents the distance from the origin to the point defined by the vector and is used to normalize vectors when finding angles.

Angle Between Two Vectors

The angle θ between two vectors can be found using the formula cos(θ) = (dot product) / (product of magnitudes). By taking the inverse cosine (arccos) of this value, we get the angle in radians or degrees, which can then be rounded as needed.