Vector A and vector B both lie in the same xy plane . Is it possible for them to have same magnitude but different components and same components but different magnitude ?
Yes, it is possible for vectors A and B to have the same magnitude but different components when they lie in the same xy plane. The magnitude of a vector is determined by its length or size, and if vectors A and B have the same magnitude, it means that they have the same length.
For example, if vector A has components A_x and A_y, and vector B has components B_x and B_y, it is possible for A_x to be different from B_x and A_y to be different from B_y, while still having the same magnitude. This can occur when the components have opposite signs or when they are arranged to cancel each other out.
On the other hand, it is not possible for vectors A and B to have the same components but different magnitudes if they lie in the same xy plane. If their components are the same, it means that their lengths or magnitudes would also be the same.
Yes, it is possible for vectors A and B to have the same magnitude but different components, and it is also possible for them to have the same components but different magnitude.
To explain this, let's first understand what components and magnitude of a vector mean.
Components of a vector refer to the projections of the vector onto the coordinate axes (x and y in this case). So, for a vector A = (Ax, Ay), Ax represents the x-component of the vector, and Ay represents the y-component of the vector.
Magnitude of a vector refers to the length or size of the vector. It is calculated using the Pythagorean theorem as the square root of the sum of the squares of its components. Therefore, the magnitude of vector A with components (Ax, Ay) is given by |A| = sqrt(Ax^2 + Ay^2).
Now, let's consider the scenario where vectors A and B have the same magnitude but different components. This means that |A| = |B|, but (Ax, Ay) ≠ (Bx, By). This is certainly possible because the components of a vector determine its direction, while the magnitude only represents its length. So, even if their components are different, they can still have the same length.
On the other hand, let's consider the scenario where vectors A and B have the same components but different magnitudes. This means that (Ax, Ay) = (Bx, By), but |A| ≠ |B|. This is also possible because the magnitude of a vector depends on the lengths of its components, so by changing the length of one or both components, you can change the magnitude of the vector without affecting the direction.
In summary, vectors can have the same magnitude but different components, as well as the same components but different magnitudes.