Mathematics 10 — Exercise 6.1 · Q6 (iii)

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.

   $$\left(\frac{1}{6}\times \sqrt{6}\right)\times î+\left(\frac{1}{6}\times \sqrt{6}\right)\times ĵ$$

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

   $$\left[\begin{array}{c}\frac{1}{6}\times \sqrt{6}\\ \frac{1}{6}\times \sqrt{6}\end{array}\right]$$

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

   $$\frac{1\times 1}{1\times 6}$$

4. Calculate these two products separately.

   $$\frac{1}{6}$$

5. Calculate this operation before combining the remaining terms.

   $$1\times \frac{1}{6}=\frac{1}{6}$$

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

   $$\mathrm{LCD}=6$$

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

   $$\frac{1}{6}=\frac{1\times 1}{6\times 1}$$

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

   $$\frac{1+1}{6}=\frac{2}{6}$$

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

   $$\mathrm{GCD}=2$$

10. Show the two divisions explicitly.

    $$\frac{\frac{2}{2}}{\frac{6}{2}}$$

11. Evaluate the numerator division.

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

12. Evaluate the denominator division.

    $$\frac{6}{2}=3$$

13. Write the reduced fraction.

    $$\frac{1}{3}$$

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

    $$\sqrt{{\left(\frac{1}{6}\times \sqrt{6}\right)}^2+{\left(\frac{1}{6}\times \sqrt{6}\right)}^2}$$

15. Square the horizontal component, including its sign.

    $${\left(\frac{1}{6}\times \sqrt{6}\right)}^2=\frac{1}{6}$$

16. Square the vertical component, including its sign.

    $${\left(\frac{1}{6}\times \sqrt{6}\right)}^2=\frac{1}{6}$$

17. Add these two squares.

    $$\frac{1}{6}+\frac{1}{6}=\frac{1}{3}$$

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

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

19. Calculate this operation before combining the remaining terms.

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

20. Take and simplify the positive square root.

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

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

    $$\frac{\left(\frac{1}{6}\times \sqrt{6}\right)\times î+\left(\frac{1}{6}\times \sqrt{6}\right)\times ĵ}{\frac{1}{3}\times \sqrt{3}}$$

22. Calculate this operation before combining the remaining terms.

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

23. Calculate this operation before combining the remaining terms.

    $$\frac{1}{6}\times \sqrt{6}\times \sqrt{3}=\frac{1}{2}\times \sqrt{2}$$

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

    $$\frac{\frac{1}{6}\times \sqrt{6}}{\frac{1}{3}\times \sqrt{3}}=\frac{1}{2}\times \sqrt{2}$$

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

    $$\frac{\frac{1}{6}\times \sqrt{6}}{\frac{1}{3}\times \sqrt{3}}=\frac{1}{2}\times \sqrt{2}$$

26. Combine both divided components.

    $$\left(\frac{1}{2}\times \sqrt{2}\right)\times î+\left(\frac{1}{2}\times \sqrt{2}\right)\times ĵ$$

Answer

The final answer is shown below.