(i) Explain why two non-zero vectors in three-dimensional space are parallel if and only if one is a scalar multiple of the other, and show how this criterion is equivalent to the condition that their vector (cross) product is the zero vector.
(ii) Let a=3i^+6j^−9k^ and b=−2i^−4j^+6k^. Verify that the two tests mentioned in part (i) both confirm that a and b are parallel.
a. Explain the equivalence of the scalar–multiple test and the zero cross-product test for parallelism of two non-zero vectors.
b. Using the vectors a=3i^+6j^−9k^ and b=−2i^−4j^+6k^:
(a) Show that one vector is a scalar multiple of the other.
(b) Compute a×b and state the result.