Mathematics 10 — Exercise Review · Q4 (ii)

Full question

For the following vectors, find 8a+b+5c.

Step-by-step solution

1. Multiply every matrix entry by the scalar.

   $$(8,\left[\begin{array}{c}2\\ -1\end{array}\right])$$

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

   $$8\times 2=16$$

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

   $$8\times \left(-1\right)=-8$$

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

   $$\left[\begin{array}{c}16\\ -8\end{array}\right]$$

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

   $$(\left[\begin{array}{c}16\\ -8\end{array}\right],\left[\begin{array}{c}3\\ 1\end{array}\right])$$

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

   $$16+3=19$$

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

   $$-8+1=-7$$

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

   $$\left[\begin{array}{c}19\\ -7\end{array}\right]$$

9. Multiply every matrix entry by the scalar.

   $$(5,\left[\begin{array}{c}4\\ 1\end{array}\right])$$

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

    $$5\times 4=20$$

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

    $$5\times 1=5$$

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

    $$\left[\begin{array}{c}20\\ 5\end{array}\right]$$

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

    $$(\left[\begin{array}{c}19\\ -7\end{array}\right],\left[\begin{array}{c}20\\ 5\end{array}\right])$$

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

    $$19+20=39$$

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

    $$-7+5=-2$$

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

    $$\left[\begin{array}{c}39\\ -2\end{array}\right]$$

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

    $$39\times î-2\times ĵ$$

Answer

The final answer is shown below.