Mathematics 10 — Exercise 6.2 · Q1 (iii)

Full question

For the following vectors, find 3a+(1/2)b.

Step-by-step solution

1. Multiply every matrix entry by the scalar.

   $$(3,\left[\begin{array}{c}7\\ -3\end{array}\right])$$

2. Entry (1,1): multiply its value by the scalar.

   $$3\times 7=21$$

3. Entry (2,1): multiply its value by the scalar.

   $$3\times \left(-3\right)=-9$$

4. Place the calculated entries in their original row and column positions.

   $$\left[\begin{array}{c}21\\ -9\end{array}\right]$$

5. Multiply every matrix entry by the scalar.

   $$(\frac{1}{2},\left[\begin{array}{c}1\\ 5\end{array}\right])$$

6. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{1\times 1}{1\times 2}$$

7. Calculate these two products separately.

   $$\frac{1}{2}$$

8. Calculate this operation before combining the remaining terms.

   $$1\times \frac{1}{2}=\frac{1}{2}$$

9. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{1\times 1}{2\times 1}$$

10. Entry (1,1): multiply its value by the scalar.

    $$\frac{1}{2}\times 1=\frac{1}{2}$$

11. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 5}{2\times 1}$$

12. Calculate these two products separately.

    $$\frac{5}{2}$$

13. Entry (2,1): multiply its value by the scalar.

    $$\frac{1}{2}\times 5=\frac{5}{2}$$

14. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{c}\frac{1}{2}\\ \frac{5}{2}\end{array}\right]$$

15. The orders match. Combine only entries in the same row and column.

    $$(\left[\begin{array}{c}21\\ -9\end{array}\right],\left[\begin{array}{c}\frac{1}{2}\\ \frac{5}{2}\end{array}\right])$$

16. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=2$$

17. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$21\times \frac{1}{1}=\frac{21\times 2}{1\times 2}$$

18. Calculate the converted numerator; the denominator is now common.

    $$\frac{42}{2}$$

19. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1}{2}=\frac{1\times 1}{2\times 1}$$

20. Keep the common denominator and add the signed numerators.

    $$\frac{42+1}{2}=\frac{43}{2}$$

21. Entry (1,1): add the corresponding entries.

    $$21+\frac{1}{2}=\frac{43}{2}$$

22. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{5\times 1}{1\times 2}$$

23. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{2}=\frac{5}{2}$$

24. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-9\times \frac{1}{1}=\frac{\left(-9\right)\times 2}{1\times 2}$$

25. Calculate the converted numerator; the denominator is now common.

    $$-\frac{18}{2}$$

26. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{5}{2}=\frac{5\times 1}{2\times 1}$$

27. Keep the common denominator and add the signed numerators.

    $$\frac{-18+5}{2}=-\frac{13}{2}$$

28. Entry (2,1): add the corresponding entries.

    $$-9+\frac{5}{2}=-\frac{13}{2}$$

29. Place the calculated entries in their original row and column positions.

    $$\left[\begin{array}{c}\frac{43}{2}\\ -\frac{13}{2}\end{array}\right]$$

30. The first component multiplies î, the second multiplies ĵ.

    $$\frac{43\times î}{2}-\frac{13\times ĵ}{2}$$

Answer

The final answer is shown below.