3.2 Scalar multiplication of vectors in component form
In Figure 27 Alice and Bob are both pushing the same face of the block of ice, but this time with the same force.
Because they are both applying the same force, we can use a single vector to represent this, say , and if the force they apply is 110 N, then
Now, the combined force exerted by both Alice and Bob is
This confirms that when we multiply a vector with a scalar quantity, the magnitude of the vector is multiplied by the scalar; if the scalar is positive its direction stays the same, but if the scalar is negative the direction is reversed.
For the situation of Alice and Bob pushing the block of ice we have
Other examples are illustrated in Figure 28. The vector can be written in component form as , and the scalar multiples of are written in component form as follows:
Scalar multiplication of a vector in component form
If and is a scalar, then
In column notation, if and is a scalar, then
For example, if , then .
Let and . Find each of the following scalar multiples.