Mathematics 10 — Exercise 6.2 · Q2 (ii)

Full question

Find the magnitude of a−2b+c.

Step-by-step solution

1. Multiply every matrix entry by the scalar.

   $$(2,\left[\begin{array}{c}1\\ 5\end{array}\right])$$

2. Entry (1,1): multiply its value by the scalar.

   $$2\times 1=2$$

3. Entry (2,1): multiply its value by the scalar.

   $$2\times 5=10$$

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

   $$\left[\begin{array}{c}2\\ 10\end{array}\right]$$

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

   $$(\left[\begin{array}{c}6\\ -1\end{array}\right],\left[\begin{array}{c}2\\ 10\end{array}\right])$$

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

   $$6-2=4$$

7. Calculate this operation before combining the remaining terms.

   $$\left(-1\right)\times 10=-10$$

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

   $$-1-10=-11$$

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

   $$\left[\begin{array}{c}4\\ -11\end{array}\right]$$

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

    $$(\left[\begin{array}{c}4\\ -11\end{array}\right],\left[\begin{array}{c}3\\ 5\end{array}\right])$$

11. Entry (1,1): add the corresponding entries.

    $$4+3=7$$

12. Entry (2,1): add the corresponding entries.

    $$-11+5=-6$$

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

    $$\left[\begin{array}{c}7\\ -6\end{array}\right]$$

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

    $$7\times î-6\times ĵ$$

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

    $$\left[\begin{array}{c}7\\ -6\end{array}\right]$$

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

    $$\sqrt{7^2+{\left(-6\right)}^2}$$

17. Square the horizontal component, including its sign.

    $$7^2=49$$

18. Square the vertical component, including its sign.

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

19. Add these two squares.

    $$49+36=85$$

20. Take and simplify the positive square root.

    $$\sqrt{85}=\sqrt{85}$$

Answer

The final answer is shown below.