1. Warm Up 1. What are the two ways that we can find the magnitude?
2. What do we use to solve for the direction?
Why don’t we use the Law of Sines or Cosines to
find the direction?

2. Objectives
Use vectors and vector addition to solve real world problems.

3. The ______________ of a vector is its length.
magnitude
When a vector is used to represent speed in a given direction, the magnitude of the vector equals the speed.

4. Ex: Find the vector sum U<1, 2> V<0, 6>
The resultant vector is the vector that represents the sum of two given vectors.
To add vectors numerically, add their components.
=<x1 + x2, y1 + y2>.
Ex: Find the vector sum
U<1, 2> V<0, 6>

5. Vector Application Steps
Step 1: Sketch the two vectors from the given information A. vector B. current
Step 2: Write the 1st vector A in component form <x,y> sketch triangle solve trig
Step 3: Write the 2nd vector B in component form
<x,y> hint: on an axis (no triangle)
Step 4: Find and sketch the resultant vector (component form)--> Vector addition
=<x1 + x2, y1 + y2>.
Step 5: Find magnitude and direction of the resultant vector
Mag distance/Pythagorean thm
Direction inverse trig then subtract from 90

6. Example 5: Aviation Application
An airplane is flying at a constant speed of 400 mi/h at a bearing of N 70º E. A 60 mi/h wind is blowing due north. What are the plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree.
Step 1 Sketch vectors for the airplane and the wind.
70°
400
Airplane
20° y x S W E N 60
Wind

7. Example 5 Continued
Step 2 Write the vector for the airplane in component form. The airplane’s vector has a magnitude of 400 mi/h and makes an angle of 20º with the x-axis.
so x = 400 cos20° 
so y = 400 sin20° 
The airplane’s vector is <375.9, 136.8>.

8. Example 5 Continued
Step 3 Write the vector for the wind in component form. Since the wind moves 60 mi/h in the direction of the y-axis, it has a horizontal component of 0 and a vertical component of 60. So its vector is <0, 60>.
Step 4 Find and sketch the resultant vector Add the components of the airplane’s vector and the wind vector.
<375.9, 136.8> + <0, 60> = <375.9, 196.8>.
The resultant vector in component form is <375.9, 196.8>.

9. Example 5 Continued
Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the airplane’s actual speed (or ground speed). +
The angle measure formed by the resultant vector gives the airplane’s actual direction.

10. Step 1 Sketch vectors for the kayaker and the current.
Example 6
What if…? Suppose the kayaker paddles at 4 mi/h at a bearing of N 20° E. There is a 1 mi/h current moving due east. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree.
20°
70°
Step 1 Sketch vectors for the kayaker and the current.

11. Check It Out! Example 6 Continued
Step 2 Write the vector for the kayaker in component form. The kayaker’s vector has a magnitude of 4 mi/h and makes an angle of 20° with the x-axis.
so x = 4cos70°  1.4.
so y = 4 sin70°  3.8.
The kayaker’s vector is <1.4, 3.8>.

12. Check It Out! Example 6 Continued
Step 3 Write the vector for the current in component form. Since the current moves 1 mi/h in the direction of the x-axis, it has a horizontal component of 1 and a vertical component of 0. So its vector is <1, 0>.
Step 4 Find and sketch the resultant vector Add the components of the kayaker’s vector and the current’s vector. <1.4, 3.8> + <1, 0> = <2.4, 3.8> The resultant vector in component form is <2.4, 3.8>.

13. Check It Out! Example 6 Continued
Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the kayak’s actual speed.
The angle measure formed by the resultant vector gives the kayak’s actual direction.
or N 32° E

14. Example 7:
A boat is heading due east at a constant speed of 35 mi/h. There is an 8 mi/h current moving north. What is the boat’s actual speed and direction?
35.9 mi/h; N 77° E