Find the angle between each pair of vectors. Round to two decimal places as necessary.
3i + 4j, j

Verified step by step guidance

1. Identify the given vectors: \( \mathbf{a} = 3\mathbf{i} + 4\mathbf{j} \) and \( \mathbf{b} = \mathbf{j} \).
2. Use the dot product formula: \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \), where \( \theta \) is the angle between the vectors.
3. Calculate the dot product: \( \mathbf{a} \cdot \mathbf{b} = (3\mathbf{i} + 4\mathbf{j}) \cdot \mathbf{j} = 0 \cdot 1 + 4 \cdot 1 = 4 \).
4. Find the magnitudes of the vectors: \( |\mathbf{a}| = \sqrt{3^2 + 4^2} = 5 \) and \( |\mathbf{b}| = \sqrt{0^2 + 1^2} = 1 \).
5. Solve for \( \cos(\theta) \) using the dot product formula: \( \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} = \frac{4}{5 \cdot 1} = \frac{4}{5} \), then find \( \theta \) using the inverse cosine function.

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) for a two-dimensional vector A = (Ax, Ay). 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.

Angle Between Vectors

The angle between two vectors can be found using the relationship cos(θ) = (A·B) / (|A| |B|). This formula allows us to derive the angle θ by taking the inverse cosine (arccos) of the dot product divided by the product of the magnitudes of the vectors. This concept is fundamental in vector analysis and applications in physics and engineering.