Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈4, 0〉, 〈2, 2〉

검증된 단계별 안내

Identify the two vectors given: \( \mathbf{u} = \langle 4, 0 \rangle \) and \( \mathbf{v} = \langle 2, 2 \rangle \).

Recall the formula for the angle \( \theta \) between two vectors \( \mathbf{u} \) and \( \mathbf{v} \): \[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \] where \( \mathbf{u} \cdot \mathbf{v} \) is the dot product and \( \|\mathbf{u}\| \) and \( \|\mathbf{v}\| \) are the magnitudes of the vectors.

Calculate the dot product: \[ \mathbf{u} \cdot \mathbf{v} = (4)(2) + (0)(2) \] Calculate the magnitudes: \[ \|\mathbf{u}\| = \sqrt{4^2 + 0^2} \] \[ \|\mathbf{v}\| = \sqrt{2^2 + 2^2} \]

Substitute the dot product and magnitudes into the cosine formula: \[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \]

Find the angle \( \theta \) by taking the inverse cosine (arccos) of the value obtained: \[ \theta = \cos^{-1} \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \right) \] Then round the result to two decimal places as required.

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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〉, it is calculated as ac + bd. This product is essential for finding the angle between vectors.

Magnitude of a Vector

The magnitude (or length) of a vector 〈x, y〉 is found using the formula √(x² + y²). It represents the distance from the origin to the point (x, y) in the plane and is used to normalize vectors when calculating 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 ratio, we obtain the angle in radians or degrees, which can then be rounded as required.

Find the angle between each pair of vectors. Round to two decimal places as necessary.〈4, 0〉, 〈2, 2〉