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Introducing vectors for engineering applications
Introducing vectors for engineering applications

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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.

Described image
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 bold v, and if the force they apply is 110 N, then

v equals 110 times i full stop

Now, the combined force exerted by both Alice and Bob is

equation sequence part 1 v plus v equals part 2 110 times i plus 110 times i equals part 3 220 times i full stop

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

two times v equals left parenthesis two multiplication 110 right parenthesis times i full stop

Other examples are illustrated in Figure 28. The vector bold a can be written in component form as a equals a sub one times i plus a sub two times j, and the scalar multiples of bold a are written in component form as follows:

equation sequence part 1 two times a equals part 2 two times left parenthesis a sub one times i plus a sub two times j right parenthesis equals part 3 two times a sub one times i plus two times a sub two times j comma
equation sequence part 1 one divided by two times a equals part 2 one divided by two times left parenthesis a sub one times i plus a sub two times j right parenthesis equals part 3 one divided by two times a sub one times i plus one divided by two times a sub two times j comma
equation sequence part 1 negative two times a equals part 2 negative two times left parenthesis a sub one times i plus a sub two times j right parenthesis equals part 3 negative two times a sub one times i minus two times a sub two times j

and

equation sequence part 1 negative one divided by two times a equals part 2 negative one divided by two times left parenthesis a sub one times i plus a sub two times j right parenthesis equals part 3 negative one divided by two times a sub one times i minus one divided by two times a sub two times j full stop
Described image
Figure 28 Scalar multiplication of a vector bold a, in component form

Scalar multiplication of a vector in component form

If a equals a sub one times i plus a sub two times j and m is a scalar, then

m times a equals m times a sub one times i plus m times a sub two times j full stop

In column notation, if a equals vector element 1 a sub one element 2 a sub two and m is a scalar, then

m times a equals vector element 1 m times a sub one element 2 m times a sub two full stop

For example, if v equals four times i minus five times j, then equation sequence part 1 three times v equals part 2 three times left parenthesis four times i minus five times j right parenthesis equals part 3 12 times i minus 15 times j.

Activity 15

Let a equals vector element 1 two element 2 negative one and b equals i plus three times j. Find each of the following scalar multiples.

  • a.four times a

  • b.negative two times a

  • c.one divided by two times a

  • d.three times b

  • e.negative four times b

  • f.one divided by three times b