Question

Vector A is 2.80 cm long and is 60.0° above the x-axis in the first quadrant. Vector B is 1.90 cm long and is 60.0° below the x-axis in the fourth quadrant. Use components to find the magnitude and direction of

a) A⋅B
b) A×B
c) B⋅A
d) B×A

Answer 1

The magnitude of A⋅B is 2.66 cm² and the magnitude of A×B is 2.45 cm². Both are scalar quantities.

Step-by-step explanation:

To find the magnitude and direction of the dot product (A⋅B), we can use the formula A⋅B = |A||B|cosθ, where θ is the angle between the two vectors. In this case, A = 2.80 cm and B = 1.90 cm, and the angle between them is 60.0°. Plugging these values into the formula, we get: A⋅B = (2.80 cm)(1.90 cm)cos(60.0°) = 2.66 cm². To find the magnitude and direction of the cross product (A×B), we can use the formula A×B = |A||B|sinθ, where θ is the angle between the two vectors. In this case, A = 2.80 cm and B = 1.90 cm, and the angle between them is 60.0°. Plugging these values into the formula, we get: A×B = (2.80 cm)(1.90 cm)sin(60.0°) = 2.45 cm².

The dot product (A⋅B) and cross product (A×B) are both scalar quantities, so they don't have a direction per se. The dot product represents the magnitude of the projection of one vector onto the other, while the cross product represents the area of the parallelogram formed by the two vectors.