Question

Which of the following is true about multiplication of a vector by a scalar.

1) Scalar multiplication by a positive number other than 1 changes its magnitude but not direction.

2) Scalar multiplication always lead to change in magnitude and direction.

3) Scalar multiplication by -1 will not change the magnitude of the vector but will change its direction.

4) Scalar multiplication by a negative number other than -1 will reverse its direction and change its magnitude as well.

• A. All are correct
• B. 1 and 4 are correct
• C. 1, 3 and 4 are correct
• D. 2, 3 and 4 are correct

Answer 1

The correct option is C 1, 3 and 4 are correct

If we multiply a vector by a scalar which is positive it will change the magnitude of the vector only and not the direction. If that scalar is less than 1, the magnitude is decreased and if it is greater than 1, then the magnitude is increased. To change the direction of the vector the sign of the vector has to be reversed. And how can we do that? By multiplying it with a negative scalar and not the positive one. So 1 is true.
Scalar multiplication will not lead to change in magnitude if the scalar is 1. (Multiplying with one doesn’t change anything) Also since it’s a positive number, so there won’t be any change in the direction, Multiplying with any positive number changes only magnitude and not direction. So 2 is false.
Scalar multiplication by -1 will not have any effect on magnitude but the minus sign will reverse the direction of the vector. So 3 is true.
Scalar multiplication by a negative number other than -1 will reverse the direction for sure because of the minus sign. Also since it’s not 1 so it will change the magnitude as well. So 4 is true.