A bar b bar are two distinct vectors that are not parallel then c =axb
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They might have meant "anti-parallel" (i.e., along the same line but in the opposite direction). The only other way to have a zero cross product is if one of the magnitudes was zero to begin with. If at least one of the vectors is zero, that would technically also make the vectors perpendicular, maybe, depending on your definitions; the dot product would certainly be zero, which is one common test for whether two vectors are perpendicular. However, for non-zero vectors that are perpendicular to each other, the cross product will not be zero.
To be concrete:
|A⃗ ×B⃗ |=|A⃗ ||B⃗ ||sin(θ)||A→×B→|=|A→||B→||sin(θ)|
where θθ is the angle between the vectors. Therefore, a zero cross product implies that one (or more) of the three terms on the right is zero: either one of the original vector magnitudes, or the sine of the angle between them. In the latter case, this only works for 0° (parallel) or 180° (anti-parallel).
To be concrete:
|A⃗ ×B⃗ |=|A⃗ ||B⃗ ||sin(θ)||A→×B→|=|A→||B→||sin(θ)|
where θθ is the angle between the vectors. Therefore, a zero cross product implies that one (or more) of the three terms on the right is zero: either one of the original vector magnitudes, or the sine of the angle between them. In the latter case, this only works for 0° (parallel) or 180° (anti-parallel).
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