Q.Let $A$, $B$, $C$ and $D$ be four points with position vectors $\bar{a} + 2\bar{b}$, $2\bar{a} - \bar{b}$, $\bar{a}$ and $3\bar{a} + \bar{b}$ respectively. Express the vectors $\overline{AC}$, $\overline{DA}$, $\overline{BA}$ and $\overline{BC}$ in terms of $\bar{a}$ and $\bar{b}$.

Vector Displacement

Displacement is the straight-line change in position from a starting point to an ending point — captured as a single vector that carries both how far and in which direction. Unlike distance travelled (which counts every wiggle of the path), displacement cares only about where you began and where you ended.

From Two Points

Suppose a particle moves from point $A$ to point $B$. Its displacement is the vector $\overrightarrow{AB}$ drawn from $A$ (tail) to $B$ (head). If $A$ and $B$ have position vectors $\vec{a}$ and $\vec{b}$ (measured from the origin), then travelling from the origin to $A$ and then along the displacement must land you at $B$: $\vec{a}+\overrightarrow{AB}=\vec{b}$. Rearranging:

A handy memory aid: "head minus tail."

Its Magnitude Is the Distance

The length of the displacement vector is the straight-line distance between the two points:

In coordinates, if $A=(x_1,y_1,z_1)$ and $B=(x_2,y_2,z_2)$, then

and its magnitude is the familiar distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$.

Displacement vs. Distance

Quick Example

A point moves from $A(1, -2, 3)$ to $B(4, 0, -1)$. …