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# Vector

- Quantities with
*Direction*- Angle
- Can be larger than 90Ė

*Magnitude*- $|\vec{V}|$
- Always positive
- A minus sign tells us about its direction

- Vectors have
*components*not*coordinates*

## Unit Vectors

- i hat
- $x$ component

- j hat
- $y$ component

## Adding Vectors

- Resultant displacement vector ($\vec{D}_R$) is the sum of the vectors
- A vector equation
- $\vec{D}_R = \vec{D}_1 + \vec{D}_2$

- The resultant vector is not equal to the sum of the magnitudes of two separate vectors, it is smaller than their sum
- $\vec{D}_R \leq \vec{D}_1 + \vec{D}_2$

- Adding vectors is
*commutative*- $\vec{V}_1 + \vec{V}_2 = \vec{V}_2 + \vec{V}_1$

$\vec{A} + \vec{B} = \begin{bmatrix}Ax\\Ay\end{bmatrix} + \begin{bmatrix}Bx\\By\end{bmatrix} = \begin{bmatrix}Ax + Bx\\Ay + By\end{bmatrix}$

### Tail-to-Tip Method

- Draw on vector ($\vec{D}_1$) to scale
- Draw second vector ($\vec{D}_2$) to scale, placing its tail at the tip of the first vector
- Make sure its direction is correct!

- Draw an arrow from the tail of $\vec{D}_1$ to the tip of $\vec{D}_2$
- This arrow represents the sum (
*resultant*) of the two vectors - The length of the resultant represents its magnitude

- This arrow represents the sum (