## Question 2: If a vector is multiplied by a positive scalar, how the result is related to the original vector? How if the scalar is zero? Negative?

### Multiplication of vector by a scalar

We know vectors are multiplied with one another. However, vectors can also be multiplied by a scalar. For example, when we find the momentum of a moving object, we multiply its mass with the velocity. Needless to say that velocity is a vector quantity and mass is a scalar.

A vector is multiplied by a scalar by multiplying the magnitude of the vector by the scalar and its direction depends upon the sign of the scalar. Thus the product of a scalar with a vector is a vector quantity. Similarly, the result (the product vector) is also related with the original vector.

i.

#### Vector multiplied by a positive scalar

When we multiply the vector by a positive scalar, the product is also a vector whose magnitude is the scalar times the magnitude of the original vector and its direction bears no change. In the following diagram, a force vector of 5 N is multiplied by 3. The product vector is 15 N in the same direction. ii.

#### Vector multiplied by negative scalar

When we multiply the vector by a negative scalar, the product is also a vector whose magnitude is the scalar times the magnitude of the original vector and its direction is opposite to the original vector (direction is reversed). In the following diagram, a displacement of 2 m is multiplied by -2. iii.

#### Vector multiplied by zero

When we multiply a vector by zero, the product vector is a null vector with zero magnitude and arbitrary direction in space.

A null vector has zero magnitude and directionless. Examples are pulling a rope from both ends in opposite directions and equal force, or a displacement from a certain point and back.