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Question

If either a = 0 or b = 0, then a×b=0. Is the converse true? Justify your answer with an example.

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Solution

If a = 0 or b = 0, then surely a×b=0
However, the converse is not true i.e., if a×b=0, then a = 0 or b = 0 may not hold.
We know a×b=ab sin θ ^n, if a and b are parallel, then θ=0 and sin 0=0.
a×b=0
As an example, consider the vectors a=^i,b=^i, then a0 and also b0.
But a×b=^i×^i=0
Alternate method
Let a=^i+^j+^k and b=4^i+4^j+4^k
|a|=12+12+12=1+1+1=30
|b|=42+42+42=16+16+16=48=430
Now, a×b=∣ ∣ ∣^i^j^k111444∣ ∣ ∣=(44)^i(44)^j+(44)^k=0
Thus, a×b=0, when |a|0 and |b|0.

Note: If a×b=0, then it is not necessary that either |a| = 0 or |b| = 0


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