1. Vectors
A vector is a quantity which has a value (magnitude) and a direction. Examples of vectors include:
Displacement
Velocity
Acceleration
Force
Weight

2. Resultant
The resultant is the sum of the vectors. There are at least 3 ways to find the resultant.
The head-to-tail method.
The parallelogram method.
The sum of horizontal and vertical components.

3. The head-to-tail method.
Connect every vector from head-to-tail. Then connect the tail of the first vector to the head of the last and that is your resultant.
Ex. A man walks 4 meters east and then 5 meters at 45° North of East.

4. Step 1 Draw the first vector to scale. Label the vector and scale.
Scale: 1 m = 1 cm
4 m East

5. Step 2 Draw the next vector to scale. Label the vector and scale.
Scale: 1 m = 1 cm
5 45° N of E
4 m East

6. Step 3 Draw the resultant to scale. Label the vector and scale.
Scale: 1 m = 1 cm
° N of E
5 45° N of E
4 m East

7. Method 2
Take the vector given and make a parallelogram with them. Note: it only works with two vectors. Then connect the ray that bisects, from your origin corner to the ending corner.

8. Step 1 Draw the first vector to scale. Label the vector and scale.
Scale: 1 m = 1 cm
4 m East

9. Step 2 Draw the next vector to scale. Label the vector and scale.
Scale: 1 m = 1 cm
5 45° N of E
4 m East

10. Step 3
Draw the same vectors connected to the opposite previous vectors to make a parallelogram.
Scale: 1 m = 1 cm
5 45° N of E
4 m East

11. Step 4 Draw the bisecting ray Measure the angle with a protractor.
Scale: 1 m = 1 cm
5 45° N of E
° N of E
4 m East

12. Method 3 (MATH)
Take each vector and find its horizontal and vertical components. All components going up and right (North and East) are positive. All components going left and down (West and South) are negative.
Make a grid and then add all components.
Horizontals are Cos θ.
Verticals are Sin θ.
If you have a vector without an angle put in a 0 for the other component.

13. Grid Layout Horizontal Vertical + 4 m East + 5 m Cos 45° E + 0 m North
+ 5 m Cos 45° N

14. Grid Layout Horizontal Vertical + 4 m East + 5 m Cos 45° E
_________________
+4 m m
Vertical
+ 0 m North
+ 5 m Sin 45° N
_________________
+0 m m

15. Grid Layout Horizontal Vertical
+ 4 m East
+ 5 m Cos 45° N of E
_________________
+4 m m
+7.53 m East
Vertical
+ 0 m North
+ 5 m Sin 45° N of E
_________________
+0 m m
+3.53 m North
Now you have a vertical and a horizontal component. Do Pythagorean Theorem and Trig to get your resultant.

16. Pythagorean Theorem and Trig
a2 + b2 = c2
= c2
8.32 m = c

17. Pythagorean Theorem and Trig
a2 + b2 = c2
= c2
8.32 m = c
Tan-1 (y / x) = θ
Tan-1 (3.53 / 7.53) = θ
25.12° N of E = θ

18. Pythagorean Theorem and Trig
a2 + b2 = c2
= c2
8.32 m = c
Tan-1 (y / x) = θ
Tan-1 (3.53 / 7.53) = θ
25.12° N of E = θ
So your answer would be:
° N of E