Mathematics 10 — Exercise Review · Q10

Full question

An aircraft flies east at 200 km/h with wind north at 60 km/h. Find resultant velocity and magnitude.

Step-by-step solution

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

   $$(\left[\begin{array}{c}200\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 60\end{array}\right])$$

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

   $$200+0=200$$

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

   $$0+60=60$$

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

   $$\left[\begin{array}{c}200\\ 60\end{array}\right]$$

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

   $$200\times î+60\times ĵ$$

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

   $$\left[\begin{array}{c}200\\ 60\end{array}\right]$$

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

   $$43600=400\times 109$$

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

   $$\sqrt{{200}^2+{60}^2}$$

9. Square the horizontal component, including its sign.

   $${200}^2=40000$$

10. Square the vertical component, including its sign.

    $${60}^2=3600$$

11. Add these two squares.

    $$40000+3600=43600$$

12. Take and simplify the positive square root.

    $$\sqrt{43600}=20\times \sqrt{109}$$

13. The direction north of east is arctan(60/200)≈16.70°.

Answer

The final answer is shown below.