For which of the following conditions will the cross product of two vectors be zero?
a) If the angle between them is 90°.
b) If the angle between them is 0°.
c) If the angle between them is 45°.
d) If the vectors have the same magnitude.
recall that one factor in the result is the sine of the angle between the vectors.
The correct condition for the cross product of two vectors to be zero is:
a) If the angle between them is 90°.
So the answer is (a).
To determine when the cross product of two vectors will be zero, we need to consider the properties of the cross product operation.
The cross product of two vectors, say u and v, is defined as a vector that is perpendicular (or orthogonal) to both u and v. Another property of the cross product is that its magnitude is equal to the product of the magnitudes of the two vectors multiplied by the sine of the angle between them.
Now, let's analyze each option to determine when the cross product will be zero:
a) If the angle between the two vectors is 90° (perpendicular), then the sine of 90° is 1. So, if we have sin(90°) = 1, multiplied by the magnitudes of the vectors, the cross product will not be zero. Therefore, option a) is incorrect.
b) If the angle between the two vectors is 0° (parallel or collinear), then the sine of 0° is 0. In this case, when we multiply the magnitudes of the vectors by sin(0°), the cross product will indeed be zero. Therefore, option b) is correct.
c) If the angle between the two vectors is 45°, then sin(45°) = √2/2. When we multiply the magnitudes of the vectors by sin(45°), the cross product will not be zero. Therefore, option c) is incorrect.
d) If the vectors have the same magnitude, it does not imply that the angle between them is either 0° or 90°. The cross product will only be zero if the vectors are parallel or collinear. Therefore, option d) is incorrect.
In summary, the correct answer is option b) "If the angle between them is 0°."