Vector

• Quantities with
  • Direction
    • Angle
    • Can be larger than 90˚
  • 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

1. Draw on vector ($\vec{D}_1$) to scale
2. Draw second vector ($\vec{D}_2$) to scale, placing its tail at the tip of the first vector
   • Make sure its direction is correct!
3. 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

Math Physics