Mathematics 10 — Exercise 6.3 · Q11

Full question

Plane airspeed is 20î and crosswind is 50ĵ. Find resultant velocity and magnitude.

Step-by-step solution

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

   $$(\left[\begin{array}{c}20\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 50\end{array}\right])$$

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

   $$20+0=20$$

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

   $$0+50=50$$

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

   $$\left[\begin{array}{c}20\\ 50\end{array}\right]$$

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

   $$20\times î+50\times ĵ$$

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

   $$\left[\begin{array}{c}20\\ 50\end{array}\right]$$

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

   $$2900=100\times 29$$

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

   $$\sqrt{{20}^2+{50}^2}$$

9. Square the horizontal component, including its sign.

   $${20}^2=400$$

10. Square the vertical component, including its sign.

    $${50}^2=2500$$

11. Add these two squares.

    $$400+2500=2900$$

12. Take and simplify the positive square root.

    $$\sqrt{2900}=10\times \sqrt{29}$$

Answer

The final answer is shown below.