Mathematics 10 — Exercise 6.1 · Q5 (iii)

Full question

Find the magnitude of the following vector.

Step-by-step solution

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

   $$\frac{î}{2}+\frac{3\times ĵ}{2}$$

2. Use the horizontal and vertical components as the two perpendicular sides.

   $$\left[\begin{array}{c}\frac{1}{2}\\ \frac{3}{2}\end{array}\right]$$

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

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

4. Calculate these two products separately.

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

5. Calculate this operation before combining the remaining terms.

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

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

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

7. Calculate these two products separately.

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

8. Calculate this operation before combining the remaining terms.

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

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

   $$\mathrm{LCD}=4$$

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

    $$\frac{1}{4}=\frac{1\times 1}{4\times 1}$$

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

    $$\frac{1}{4}$$

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

    $$\frac{9}{4}=\frac{9\times 1}{4\times 1}$$

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

    $$\frac{9}{4}$$

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

    $$\frac{1+9}{4}=\frac{10}{4}$$

15. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=2$$

16. Show the two divisions explicitly.

    $$\frac{\frac{10}{2}}{\frac{4}{2}}$$

17. Evaluate the numerator division.

    $$\frac{10}{2}=5$$

18. Evaluate the denominator division.

    $$\frac{4}{2}=2$$

19. Write the reduced fraction.

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

20. Magnitude is the nonnegative square root of the sum of component squares.

    $$\sqrt{{\left(\frac{1}{2}\right)}^2+{\left(\frac{3}{2}\right)}^2}$$

21. Square the horizontal component, including its sign.

    $${\left(\frac{1}{2}\right)}^2=\frac{1}{4}$$

22. Square the vertical component, including its sign.

    $${\left(\frac{3}{2}\right)}^2=\frac{9}{4}$$

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

    $$\frac{1\times 1}{1\times 4}$$

24. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{4}=\frac{1}{4}$$

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

    $$\frac{9\times 1}{1\times 4}$$

26. Calculate this operation before combining the remaining terms.

    $$9\times \frac{1}{4}=\frac{9}{4}$$

27. Add these two squares.

    $$\frac{1}{4}+\frac{9}{4}=\frac{5}{2}$$

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

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

29. Calculate this operation before combining the remaining terms.

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

30. Take and simplify the positive square root.

    $$\sqrt{\frac{5}{2}}=\frac{1}{2}\times \sqrt{10}$$

Answer

The final answer is shown below.