Mathematics 10 — Exercise 6.2 · Q2 (iii)

Full question

Find the magnitude of c−b−a.

Step-by-step solution

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

   $$(\left[\begin{array}{c}3\\ 5\end{array}\right],\left[\begin{array}{c}1\\ 5\end{array}\right])$$

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

   $$3-1=2$$

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

   $$5-5=0$$

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

   $$\left[\begin{array}{c}2\\ 0\end{array}\right]$$

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

   $$(\left[\begin{array}{c}2\\ 0\end{array}\right],\left[\begin{array}{c}6\\ -1\end{array}\right])$$

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

   $$2-6=-4$$

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

   $$0+1=1$$

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

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

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

   $$-4\times î+ĵ$$

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

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

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

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

12. Square the horizontal component, including its sign.

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

13. Square the vertical component, including its sign.

    $$1^2=1$$

14. Add these two squares.

    $$16+1=17$$

15. Take and simplify the positive square root.

    $$\sqrt{17}=\sqrt{17}$$

Answer

The final answer is shown below.