What can be said about two vectors whose cross product is the zero vector?
they are parallel
since the cross product involves the cosine function. Cos(180) = 0 :)
*** cos(90) = 0 my mistake
Good lord.
xross product is a sine function, if it is zero, the angle is zero or 180, so they are in the same or opposite directions.
When the cross product of two vectors is the zero vector, it indicates a specific relationship between the two vectors called "collinearity" or "parallelism". Here's an explanation of how to determine the relationship between two vectors using their cross product:
1. Cross Product Calculation:
- Given two vectors, let's say vector A = [A1, A2, A3] and vector B = [B1, B2, B3].
- Calculate the cross product of the two vectors using the formula:
A × B = [(A2 * B3) - (A3 * B2), (A3 * B1) - (A1 * B3), (A1 * B2) - (A2 * B1)].
2. Interpreting the Result:
- If the resulting cross product is the zero vector [0, 0, 0], it implies that the two vectors are parallel or collinear (lie on the same line).
- In other words, the vectors A and B have the same or opposite direction.
3. Parallel Vectors:
- If the two vectors are parallel, they share the same direction.
- For example, if vector A is [1, 2, 3] and vector B is [2, 4, 6], the cross product will be [0, 0, 0], indicating that they are parallel.
4. Opposite Direction Vectors:
- If the two vectors have opposite directions, they are still collinear but pointing in opposite directions.
- For example, if vector A is [1, 2, 3] and vector B is [-2, -4, -6], the cross product will still be [0, 0, 0], showing that they are collinear.
To summarize, if the cross product of two vectors is the zero vector, it means that the vectors are either parallel or collinear, and have the same or opposite directions.