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Introducing vectors for engineering applications
Introducing vectors for engineering applications

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3.4 Combining vector operations

When using vectors to model engineering systems it is often necessary to carry out multiple operations to combine vectors in different ways. When vectors are expressed in component form, combining operations involves following the standard rules of algebra. In the next example, the vector operations of addition, subtraction and scalar multiplication are combined.

Example 5 Simplifying a combination of vectors in component form

Let t equals five times i plus three times j , u equals negative two times i plus seven times j and v equals four times i minus four times j .

Find three times t minus u minus five times v in component form.

Solution

Substitute in the expressions for bold t , bold u and bold v in terms of bold i and bold j , and then simplify:

equation sequence part 1 three times t minus u minus five times v equals part 2 three times left parenthesis five times i plus three times j right parenthesis minus left parenthesis negative two times i plus seven times j right parenthesis minus five times left parenthesis four times i minus four times j right parenthesis equals part 3 sum with 3 summands 15 times i plus nine times j plus two times i minus seven times j minus 20 times i plus 20 times j equals part 4 15 times i plus two times i minus 20 times i plus nine times j minus seven times j plus 20 times j equals part 5 negative three times i plus 22 times j full stop

Activity 17

Find each of the following vectors in component form.

  • a. sum with 3 summands negative two times a plus three times b plus four times c , where a equals two times i plus three times j , b equals i minus four times j and c equals negative five times i plus seven times j

  • b. two times vector element 1 six element 2 negative three minus seven times vector element 1 one element 2 two plus five times vector element 1 negative one element 2 four

  • c. a sub one times vector element 1 one element 2 zero plus a sub two times vector element 1 zero element 2 one , where a sub one and a sub two are any scalars