Mathematics 10 — Exercise 6.1 · Q6 (iv)

Full question

Find a unit vector in the direction of the following vector.

Step-by-step solution

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

   $$\frac{î}{2}-\frac{3\times ĵ}{4}$$

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

   $$\left[\begin{array}{c}\frac{1}{2}\\ -\frac{3}{4}\end{array}\right]$$

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.

   $$\frac{\left(-3\right)\times 1}{1\times 4}$$

7. Calculate these two products separately.

   $$-\frac{3}{4}$$

8. Calculate this operation before combining the remaining terms.

   $$\left(-3\right)\times \frac{1}{4}=-\frac{3}{4}$$

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

   $$\mathrm{LCD}=16$$

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

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

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

    $$\frac{4}{16}$$

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

    $$\frac{9}{16}=\frac{9\times 1}{16\times 1}$$

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

    $$\frac{9}{16}$$

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

    $$\frac{4+9}{16}=\frac{13}{16}$$

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

    $$\sqrt{{\left(\frac{1}{2}\right)}^2+{\left(-\frac{3}{4}\right)}^2}$$

16. Square the horizontal component, including its sign.

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

17. Square the vertical component, including its sign.

    $${\left(-\frac{3}{4}\right)}^2=\frac{9}{16}$$

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

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

19. Calculate these two products separately.

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

20. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{9\times 1}{1\times 16}$$

22. Calculate this operation before combining the remaining terms.

    $$9\times \frac{1}{16}=\frac{9}{16}$$

23. Add these two squares.

    $$\frac{1}{4}+\frac{9}{16}=\frac{13}{16}$$

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

    $$\frac{13\times 1}{1\times 16}$$

25. Calculate these two products separately.

    $$\frac{13}{16}$$

26. Calculate this operation before combining the remaining terms.

    $$13\times \frac{1}{16}=\frac{13}{16}$$

27. Take and simplify the positive square root.

    $$\sqrt{\frac{13}{16}}=\frac{1}{4}\times \sqrt{13}$$

28. A unit vector in this direction is the original vector divided by its magnitude.

    $$\frac{\frac{î}{2}-\frac{3\times ĵ}{4}}{\frac{1}{4}\times \sqrt{13}}$$

29. Calculate this operation before combining the remaining terms.

    $$\frac{1}{\frac{1}{2}\times \sqrt{13}}=\frac{2}{13}\times \sqrt{13}$$

30. Calculate this operation before combining the remaining terms.

    $$1\times \frac{2}{13}\times \sqrt{13}=\frac{2}{13}\times \sqrt{13}$$

31. Divide this component by the same magnitude and simplify.

    $$\frac{1}{\frac{1}{2}\times \sqrt{13}}=\frac{2}{13}\times \sqrt{13}$$

32. Calculate this operation before combining the remaining terms.

    $$\frac{1}{\sqrt{13}}=\frac{1}{13}\times \sqrt{13}$$

33. Calculate this operation before combining the remaining terms.

    $$\left(-3\right)\times \frac{1}{13}\times \sqrt{13}=\left(-\frac{3}{13}\right)\times \sqrt{13}$$

34. Divide this component by the same magnitude and simplify.

    $$-\frac{3}{\sqrt{13}}=\left(-\frac{3}{13}\right)\times \sqrt{13}$$

35. Combine both divided components.

    $$\left(\frac{2}{13}\times \sqrt{13}\right)\times î+\left(\left(-\frac{3}{13}\right)\times \sqrt{13}\right)\times ĵ$$

Answer

The final answer is shown below.