Practice question · Sort into groups
Sort each pair of vectors by whether its cross product is the zero vector.
Groups: Cross product is zero · Cross product is nonzero
- (1, 0, 0) and (0, 1, 0)
- (1, 2, 3) and (2, 4, 6)
- (0, 0, 5) and (0, 0, -3)
- (2, 1, 0) and (1, 3, 0)
- (1, 1, 1) and (-2, -2, -2)
Hints
- The cross product vanishes exactly when the two vectors are parallel.
- Check whether one vector is a scalar multiple of the other, a negative multiple counts.
Show the answer
Cross product is zero: (1, 2, 3) and (2, 4, 6), (1, 1, 1) and (-2, -2, -2), (0, 0, 5) and (0, 0, -3)
Cross product is nonzero: (1, 0, 0) and (0, 1, 0), (2, 1, 0) and (1, 3, 0)
Why
(2, 4, 6) = 2(1, 2, 3), (-2, -2, -2) = -2(1, 1, 1) and (0, 0, -3) is a multiple of (0, 0, 5), so those three pairs are parallel and span no area at all. The other two pairs genuinely span a parallelogram, so their cross products are nonzero.
Practise Cross Product and Geometry in Space
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