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

Verified step by step guidance

1. Identify the two vectors given: \( \mathbf{u} = \langle 4, 0 \rangle \) and \( \mathbf{v} = \langle 2, 2 \rangle \).
2. Recall the formula for the angle \( \theta \) between two vectors \( \mathbf{u} \) and \( \mathbf{v} \): \[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \] where \( \mathbf{u} \cdot \mathbf{v} \) is the dot product and \( \|\mathbf{u}\| \) and \( \|\mathbf{v}\| \) are the magnitudes of the vectors.
3. Calculate the dot product: \[ \mathbf{u} \cdot \mathbf{v} = (4)(2) + (0)(2) \] Calculate the magnitudes: \[ \|\mathbf{u}\| = \sqrt{4^2 + 0^2} \] \[ \|\mathbf{v}\| = \sqrt{2^2 + 2^2} \]
4. Substitute the dot product and magnitudes into the cosine formula: \[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \]
5. 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.

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〉, 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.