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.

Figure 27 Alice and Bob pushing a block of ice with the same force

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Alice and Bob push on the same side of a block of ice so that it moves to the right with the same force.

Figure 27 Alice and Bob pushing a block of ice 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:

and

Maximise

Figure 28 Scalar multiplication of a vector , in component form

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(a) A vector a is made of horizontal and vertical components a1i and a2j. (b) Top: The vector a has been multiplied by a factor of 2 with components 2a1i and 2a2j. Bottom: The vector a has been multiplied by a factor of 1/2 with components 1/2a1i and 1/2a2j. (c) As in b but the factor by which the vector has been scaled is negated resulting in a change of direction.

Figure 28 Scalar multiplication of a vector MathJax failure: MathML - Unexpected text node: '

For example, if , then .