Vector Addition

Graphically, we combine vectors by placing them head-to-tail. The resulting vector (the "sum" or "resultant") goes from the tail of the first vector to the head of the last vector.

Numerically, we add the corresponding components of the vectors:

Use Cases

• Finding the resultant displacement after multiple movements.
• Calculating the net force when multiple forces act on an object.
• Determining relative velocity (e.g., the velocity of an object relative to a moving observer).

Vector Subtraction

Subtracting a vector $\vec{B}$ from $\vec{A}$ is the same as adding the negative of $\vec{B}$ (a vector with the same magnitude but opposite direction).

Numerically, we subtract the corresponding components:

Graphically, subtraction $\vec{A} - \vec{B}$ is often visualized by placing the vectors tail-to-tail. The resultant vector points from the head of $\vec{B}$ to the head of $\vec{A}$. This shows the change needed to get from $\vec{B}$ to $\vec{A}$ (since $\vec{B} + (\vec{A} - \vec{B}) = \vec{A}$).

Use Cases

• Finding the change in velocity (acceleration is related to this): $\Delta\vec{v} = \vec{v}_f - \vec{v}_i$.
• Determining relative position vectors.

Scalar Multiplication

This operation scales a vector, changing its magnitude and potentially reversing its direction.

Concept

Multiplying a vector $\vec{A}$ by a scalar (a regular number) $c$ multiplies each component by $c$:

• If $c > 0$, the vector's magnitude is scaled by $c$, and its direction remains the same.
• If $c < 0$, the vector's magnitude is scaled by $\|c\|$, and its direction is reversed (flipped 180°).
• If $c = 0$, the result is the zero vector $\langle 0, 0, 0 \rangle$.

Utility

• Relating force and acceleration via mass ($\vec{F} = m\vec{a}$).
• Relating velocity and momentum via mass ($\vec{p} = m\vec{v}$).
• Creating unit vectors: A unit vector $\hat{u}$ in the same direction as $\vec{u}$ is found by $\hat{u} = \frac{1}{\|\vec{u}\|}\vec{u}$. This involves scalar multiplication by $1/\|\vec{u}\|$.