Mathematics 10 — Exercise 6.2 · Q1 (vi)

Full question

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

Step-by-step solution

1. Multiply every matrix entry by the scalar.

   $$(\frac{3}{2},\left[\begin{array}{c}7\\ -3\end{array}\right])$$

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

   $$\frac{3\times 1}{1\times 2}$$

3. Calculate these two products separately.

   $$\frac{3}{2}$$

4. Calculate this operation before combining the remaining terms.

   $$3\times \frac{1}{2}=\frac{3}{2}$$

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

   $$\frac{3\times 7}{2\times 1}$$

6. Calculate these two products separately.

   $$\frac{21}{2}$$

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

   $$\frac{3}{2}\times 7=\frac{21}{2}$$

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

   $$\frac{3\times \left(-3\right)}{2\times 1}$$

9. Calculate these two products separately.

   $$-\frac{9}{2}$$

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

    $$\frac{3}{2}\times \left(-3\right)=-\frac{9}{2}$$

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

    $$\left[\begin{array}{c}\frac{21}{2}\\ -\frac{9}{2}\end{array}\right]$$

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

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

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

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

14. Calculate this operation before combining the remaining terms.

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

15. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 1=-1$$

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.

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

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

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

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

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

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

    $$\frac{21-2}{2}=\frac{19}{2}$$

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

    $$\frac{21}{2}-1=\frac{19}{2}$$

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

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

23. Calculate this operation before combining the remaining terms.

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

24. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 5=-5$$

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

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

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

    $$-5\times \frac{1}{1}=\frac{\left(-5\right)\times 2}{1\times 2}$$

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

    $$-\frac{10}{2}$$

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

    $$\frac{-9-10}{2}=-\frac{19}{2}$$

29. Entry (2,1): subtract the corresponding entries.

    $$-\frac{9}{2}-5=-\frac{19}{2}$$

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

    $$\left[\begin{array}{c}\frac{19}{2}\\ -\frac{19}{2}\end{array}\right]$$

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

    $$\frac{19\times î}{2}-\frac{19\times ĵ}{2}$$

Answer

The final answer is shown below.