The resultant vector is the vector that 'results' from adding two or more vectors together. There are a two different ways to calculate the resultant vector.

Methods for calculating a Resultant Vector- The head to tail method to calculate a resultant which involves lining up the head of the one vector with the tail of the other.
- The parallelogram method to calculate resultant vector. This method involves properties of parallelograms but, in the end, boils down to a simple formula.

In the picture on the left, the black vector is the resultant of the two red vectors. To try to understand what a resultant is consider the following story.

If you drove from your house, centered at the origin. To your friends house, at the point (3,4), imagine that you had to take two different roads these are the two red vectors. However, the resultant vector vector would be the straight line path from your home to your friend's house, and the black vector represents that path.

### Head to Tail Method

- Place the two vectors next to each other such that the head of the one vector is touching the tail of the other vector.
- Draw the resultant vector by starting where

### Steps for Head to Tail Method

### Calculate the magnitude resultant vector

To find the resultant vector's magnitude, use the pythagorean theorem

**Practice** Problems

How long is the vector that you drew?

### Parallelogram Method to Calculate Resultant

Before tackling the parallelogram method for solving resultant vectors, you should be comfortable with the following topics

- SOHCAHTOA (basic sine, cosine, tangent )
- law of cosines
- law of sines
- the following properties of parallelograms
- opposite sides of parallelograms are congruent
- oppositeangles of parallelograms are congruent

**Step 1)** Draw a parallelogram based on the two vectors that you already have. These vectors will be two sides of the parallelogram (not the opposite sides since they have the angle between them)

**Step 2)** We now have a parallelogram and know two angles (opposite angles of parallelograms are congruent). We can also figure out the other pair of angles since the other pair are congruent and all four angles must add up to 360.

**Step 3** Draw the paprallelograms diagonal. This diagonal is **the resultant vector**

Use the law of cosines to determine the length of the resultant

Use the law of cosines to calculate the resultant.