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Introducing vectors for engineering applications
Introducing vectors for engineering applications

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4.1 Scalar product of a vector from components

Consider vectors bold a and bold b in Figure 29. In component form these are written as a equals a sub one times i plus a sub two times j and b equals b sub one times i plus b sub two times j . How can we calculate the scalar product a dot operator b ?

Described image
Figure 29 Finding the scalar product of bold a and bold b by comparing components

The scalar product will tell us how much vector bold b will grow vector bold a , and to determine this we want to identify how much the vectors interact. One method is to consider how much the horizontal and vertical components of the vectors interact, as illustrated in Figure 30. There are four possible combinations to consider: horizontal to horizontal, horizontal to vertical, vertical to horizontal, and vertical to vertical.

Described image
Figure 30 Interacting component vectors in the scalar product of bold a and bold b

Horizontal components do not interact with vertical components (and vice versa) because they are independent of each other, so a sub one dot operator b sub two equals zero and a sub two dot operator b sub one equals zero , and they do not contribute to the value of scalar product. Horizontal components interact with each other, and vertical components interact with each other, so a sub one dot operator b sub one and a sub two dot operator b sub two both contribute to the value of  a dot operator b .

The expression a sub one dot operator b sub one is a measure of how much the scalar quantity b sub one grows the scalar quantity a sub one , so it is equal to a sub one multiplication b sub one , and similarly a sub two dot operator b sub two is equal to a sub two multiplication b sub two . The scalar product is a combination of these, so

a dot operator b equals a sub one multiplication b sub one plus a sub two multiplication b sub two full stop

For example, if a equals two times i plus three times j and b equals i minus two times j , then

equation sequence part 1 a dot operator b equals part 2 left parenthesis two multiplication one right parenthesis plus left parenthesis three multiplication left parenthesis negative two right parenthesis right parenthesis equals part 3 two minus six equals part 4 negative four full stop

Scalar product of vectors in terms of components

If a equals a sub one times i plus a sub two times j and b equals b sub one times i plus b sub two times j , or a equals vector element 1 a sub one element 2 a sub two and b equals vector element 1 b sub one element 2 b sub two in column notation, then

a dot operator b equals a sub one times b sub one plus a sub two times b sub two full stop

Activity 19

Suppose that u equals three times i plus four times j , v equals negative two times i plus three times j and w equals negative i minus j . Find the following.

  • a. u dot operator v

  • b. u dot operator w

  • c. v dot operator w