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. Step 1: Recall the formula for the angle \( \theta \) between two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \): \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} \).
2. Step 2: Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \) for vectors \( \langle 4, 0 \rangle \) and \( \langle 2, 2 \rangle \): \( 4 \times 2 + 0 \times 2 = 8 \).
3. Step 3: Find the magnitude of each vector. For \( \mathbf{a} = \langle 4, 0 \rangle \), \( \|\mathbf{a}\| = \sqrt{4^2 + 0^2} = 4 \). For \( \mathbf{b} = \langle 2, 2 \rangle \), \( \|\mathbf{b}\| = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \).
4. Step 4: Substitute the dot product and magnitudes into the cosine formula: \( \cos \theta = \frac{8}{4 \times 2\sqrt{2}} \).
5. Step 5: Solve for \( \theta \) by taking the inverse cosine (arccos) of the result from Step 4, and round the angle to two decimal places.

Key Concepts

Dot Product

The dot product of two vectors is a scalar value that is calculated by multiplying their corresponding components and summing the results. It is given by the formula A·B = Ax * Bx + Ay * By. The dot product is crucial for finding the angle between two vectors, as it relates to the cosine of the angle through the equation A·B = |A| |B| cos(θ).

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula |A| = √(Ax² + Ay²). For two-dimensional vectors, this involves taking the square root of the sum of the squares of its components. Understanding the magnitude is essential for determining the angle between vectors, as it is used in the dot product formula.

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 allows us to find the angle θ by taking the inverse cosine (arccos) of the calculated cosine value. This concept is fundamental in trigonometry for relating angles to the properties of vectors.