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

검증된 단계별 안내

Identify the two vectors given: \( \mathbf{u} = \langle 2, 1 \rangle \) and \( \mathbf{v} = \langle -3, 1 \rangle \).

Recall the formula for the angle \( \theta \) between two vectors \( \mathbf{u} \) and \( \mathbf{v} \): \[\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}\]

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.

Find the magnitudes of each vector using: \[\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}\] and \[\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}\]

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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