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. insert step 1: Understand that the angle \( \theta \) between two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) can be found using the dot product formula: \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} \).
2. insert step 2: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \) for the vectors \( \langle 2, 1 \rangle \) and \( \langle -3, 1 \rangle \) using the formula: \( a_1 \cdot b_1 + a_2 \cdot b_2 \).
3. insert step 3: Find the magnitudes of each vector. For vector \( \langle 2, 1 \rangle \), use the formula \( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2} \). Similarly, find the magnitude of vector \( \langle -3, 1 \rangle \).
4. insert step 4: Substitute the dot product and magnitudes into the formula \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} \) to find \( \cos \theta \).
5. insert step 5: Use the inverse cosine function to find the angle \( \theta \) in degrees, rounding to two decimal places as necessary.

Key Concepts

Dot Product

The dot product of two vectors is a scalar value obtained by multiplying their corresponding components and summing the results. It is calculated as A·B = A1*B1 + A2*B2 for vectors A = 〈A1, A2〉 and B = 〈B1, B2〉. The dot product is crucial for finding the angle between vectors, as it relates to the cosine of the angle through the formula A·B = |A| |B| cos(θ).

Magnitude of a Vector

The magnitude of a vector is its length, calculated using the formula |A| = √(A1² + A2²) for a vector A = 〈A1, A2〉. This value is essential for determining the angle between vectors, as it is used in the dot product formula. Understanding how to compute the magnitude allows for accurate calculations of angles and comparisons between vector lengths.

Cosine of the Angle

The cosine of the angle between two vectors can be derived from the dot product and the magnitudes of the vectors. Specifically, cos(θ) = (A·B) / (|A| |B|). This relationship is fundamental in trigonometry, as it allows us to find the angle θ by taking the inverse cosine (arccos) of the calculated value. Rounding the result to two decimal places is often required for precision in final answers.