Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈1, 6〉, 〈-1, 7〉

Verified step by step guidance

1. Calculate the dot product of the vectors \( \langle 1, 6 \rangle \) and \( \langle -1, 7 \rangle \) using the formula: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).
2. Find the magnitude of each vector. For vector \( \langle 1, 6 \rangle \), use the formula: \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \).
3. Similarly, find the magnitude of vector \( \langle -1, 7 \rangle \) using the same formula: \( \| \mathbf{b} \| = \sqrt{b_1^2 + b_2^2} \).
4. Use the dot product and magnitudes to find the cosine of the angle \( \theta \) between the vectors with the formula: \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \| \| \mathbf{b} \|} \).
5. Calculate the angle \( \theta \) by taking the inverse cosine (arccos) of the result from the previous step.

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 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). Understanding how to compute the magnitude is essential for determining the angle between vectors, as it is used in the dot product formula to normalize the vectors and isolate the cosine of the angle.

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, which is necessary for solving the problem of finding the angle between the given vectors.