The scalar product (aka dot product) of two perpendicular vectors is
The Correct Answer and Explanation is:
The scalar product (or dot product) of two perpendicular vectors is zero. Here’s why:
The dot product of two vectors A and B is defined as:A⋅B=∣A∣∣B∣cosθ\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \thetaA⋅B=∣A∣∣B∣cosθ
where:
- A\mathbf{A}A and B\mathbf{B}B are the two vectors.
- ∣A∣|\mathbf{A}|∣A∣ and ∣B∣|\mathbf{B}|∣B∣ are the magnitudes (or lengths) of the vectors.
- θ\thetaθ is the angle between the two vectors.
For two vectors to be perpendicular, the angle θ\thetaθ between them must be 90°.
Now, since cos90°=0\cos 90° = 0cos90°=0, the dot product formula becomes:A⋅B=∣A∣∣B∣⋅0=0\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cdot 0 = 0A⋅B=∣A∣∣B∣⋅0=0
This means that the scalar (dot) product of two perpendicular vectors is always zero, regardless of their magnitudes.
Why does this make sense physically?
The dot product measures the extent to which two vectors point in the same or opposite directions. When two vectors are perpendicular, they do not share any directional component with each other. Essentially, there is no “overlap” in their directionality, which is why their dot product equals zero.
This concept is useful in many fields of physics and engineering, particularly when analyzing forces, velocities, or other vector quantities. For example, if a force is applied perpendicular to the motion of an object, no work is done, which is consistent with the fact that the dot product of force and displacement is zero in this case.
In summary, the dot product of two perpendicular vectors is always zero, and this is a direct consequence of the geometric relationship between the vectors.
