Question

If vectors A and B are orthogonal, what is the component of B along the direction of A? What is the component of A along the direction of B?

Answer 1

The component of $\mathbf{B}$ along the direction of $\mathbf{A}$ is given by the scalar product:

$$B_A =\dfrac{ \mathbf{A}\cdot \mathbf{B}}{|\mathbf{\mathbf{B}}|}$$

where $|\mathbf{B}|$ is the magnitude of the vector. Since for the orthogonal vectors the scalar product is equal to 0 ($\mathbf{A}\cdot \mathbf{B} = 0$ ). Thus, the component of $\mathbf{B}$ along the direction of $\mathbf{A}$ is zero.

Similarly, the component of $\mathbf{A}$ along the direction of $\mathbf{B}$ is:

$$A_B =\dfrac{ \mathbf{B}\cdot \mathbf{A}}{|\mathbf{\mathbf{A}}|}$$

Since $\mathbf{B}\cdot \mathbf{A} = \mathbf{A}\cdot \mathbf{B} =0$ (the scalar product is commutative), the component of $\mathbf{A}$ along the direction of $\mathbf{B}$ is zero as well.

Answer. 0 and 0.