Question

Two vectors A and B lie in the xy plane.

a. Can vector A have the same magnitude as vector B but different components?
b. Can vector A have the same component as vector B but a different magnitude?​

Answer 1

a. Yes, vector A can have the same magnitude as vector B but different components. For example, vector A could point in the positive x direction with a magnitude of 5, while vector B could point in the positive y direction with a magnitude of 5. Both vectors have the same magnitude of 5, but their components are different.

b. Yes, vector A can have the same component as vector B but a different magnitude. For example, if vector B points in the positive x direction with a magnitude of 3, then vector A could also point in the positive x direction with a magnitude of 3, and have the same x component as vector B. However, vector A could have a different magnitude, such as pointing half the distance of vector B, in which case the magnitude of vector A would be 1.5

Answer 2

a. It is possible for two vectors A and B to have the same magnitude but different components in the xy plane. The magnitude of a vector is determined by the formula (A^2 + B^2)^(1/2), where A and B are the components of the vector in the x and y directions, respectively. If two vectors have the same magnitude but point in different directions, then they must have different components.

b. It is possible for two vectors A and B to have the same component in the x direction (or in the y direction), but different magnitudes. For two vectors to be considered equal, they must have the same components in both the x and y directions. Therefore, if two vectors have different magnitudes, even if they have a component in common, they are considered different vectors.

I hope this answer is clearer and helps you better understand the properties of vectors in the xy plane.