Question

Evaluate A.B and the angle between A and B:
(a) A = 3i – 4j; B = -2i + 6j

Answer 1

To evaluate the dot product of vectors A and B, calculate the sum of the products of their corresponding components, resulting in -30. To find the angle between the vectors, use the dot product, the magnitudes of A and B, and the cosine inverse function.

Step-by-step explanation:

To evaluate A.B (the dot product) and find the angle between vectors A and B, where A = 3i – 4j and B = -2i + 6j, we follow these steps:

1. Calculate the dot product: A.B = Ax Bx + Ay By which yields (3)(-2) + (-4)(6) = -6 - 24 = -30.
2. To find the angle between A and B, we use the formula: cos(θ) = (A.B) / (|A| |B|), where |A| and |B| are the magnitudes of A and B.
3. Calculate the magnitudes of A and B: |A| = √(3² + (-4)²) = 5 and |B| = √((-2)² + 6²) = √(40) = √4*√10 = 2√10.
4. Then cos(θ) = -30 / (5 * 2√10) = -3 / √10. To find the angle, we take the inverse cosine (θ = cos⁻¹(-3/√10)).

We have now found the dot product and the method for finding the angle between the two vectors.