Mathematics 10 — Exercise 6.3 · Q5

Full question

Given AB=6î+2ĵ and DC=−6î−2ĵ, show equal magnitude and parallelism.

Step-by-step solution

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

   $$\left[\begin{array}{c}6\\ 2\end{array}\right]$$

2. Extract a perfect-square factor from under the root.

   $$40=4\times 10$$

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

   $$\sqrt{6^2+2^2}$$

4. Square the horizontal component, including its sign.

   $$6^2=36$$

5. Square the vertical component, including its sign.

   $$2^2=4$$

6. Add these two squares.

   $$36+4=40$$

7. Take and simplify the positive square root.

   $$\sqrt{40}=2\times \sqrt{10}$$

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

   $$\left[\begin{array}{c}-6\\ -2\end{array}\right]$$

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

   $$\sqrt{{\left(-6\right)}^2+{\left(-2\right)}^2}$$

10. Square the horizontal component, including its sign.

    $${\left(-6\right)}^2=36$$

11. Square the vertical component, including its sign.

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

12. Add these two squares.

    $$36+4=40$$

13. Take and simplify the positive square root.

    $$\sqrt{40}=2\times \sqrt{10}$$

14. The second vector is −1 times the first, so they are parallel in opposite directions; the computed magnitudes are equal.

    $$-6\times î-2\times ĵ=-(6\times î+2\times ĵ)$$

Answer

The final answer is shown below.