Question

If either or , then . Is the converse true? Justify your answer with an example.

Answer 1

It is given that a → =0 or b → =0 , then a → × b → =0.

To discuss about the converse part, let us take an example,

Take parallel non-zero vectors, so that a → × b → =0.

Let, a → =2 i ^ +3 j ^ +4 k ^ and b → =4 i ^ +6 j ^ +8 k ^ .

a → =2

Then the cross product of two vectors ( a 1 i ^ + a 2 j ^ + a 3 k ^ ) and ( b 1 i ^ + b 2 j ^ + b 3 k ^ ) is given by,

a → × b= → | i ^ j ^ k ^ a 1 a 2 a 3 b 1 b 2 b 3 |

a → × b → =| i ^ j ^ k ^ 2 3 4 4 6 8 | = i ^ ( 24−24 )− j ^ ( 16−16 )+ k ^ ( 12−12 ) = i ^ ( 0 )− j ^ ( 0 )+ k ^ ( 0 )

Since,

| a |= 2 2 + 3 2 + 4 2 = 29

Hence, a → ≠0.

Since ,

| b → |= 4 2 + 6 2 + 8 2 = 116

Hence, b → ≠0.

Thus, if a → =0 or b → =0 , then a → × b → =0. Then the converse part need not be true.