How two vectors behave if their dot product is zero
The Correct Answer and Explanation is :
If the dot product of two vectors is zero, it means that the vectors are orthogonal (perpendicular) to each other.
Explanation:
The dot product of two vectors A and B is given by:
[
A \cdot B = |A| |B| \cos(\theta)
]
where:
- ( |A| ) and ( |B| ) are the magnitudes (lengths) of the vectors,
- ( \theta ) is the angle between them,
- ( \cos(\theta) ) is the cosine of that angle.
For the dot product to be zero:
[
|A| |B| \cos(\theta) = 0
]
Since the magnitudes of vectors cannot be zero unless they are zero vectors, the only way this equation holds is if:
[
\cos(\theta) = 0
]
This occurs when:
[
\theta = 90^\circ \text{ (or ( \frac{\pi}{2} ) radians)}
]
Thus, if the dot product of two vectors is zero, the angle between them is 90°, meaning they are perpendicular (orthogonal) to each other.
Why Is This Important?
- In Physics: Perpendicular vectors appear in force calculations, work done by a force, and electromagnetic field components.
- In Computer Graphics: Used in normal vector calculations to determine surface orientations.
- In Linear Algebra: Orthogonal vectors play a key role in vector spaces and transformations.
Below is a visual representation of two perpendicular vectors, A and B, whose dot product is zero.
The image above visually represents two perpendicular vectors A and B, originating from the same point (the origin), forming a 90-degree angle. This demonstrates that their dot product is zero, confirming their orthogonality. Let me know if you need any modifications!