Revision of Position Vectors

Video lesson · from 0:131 worked example

One lesson video covers all of Vectors, so it opens at 0:13 for this topic — not from the beginning.

The vector from origin $O$ to point $A$ is called the position vector of point A: $\vec{OA}$ or $\vec{a}$ .

The vector $\vec{AB}$ is the vector which originates at A and ends at B. $\vec{AB}$ is the position vector of B relative to A.

\vec{AB} = \vec{b} - \vec{a}

$\vec{AB} = \mathbf{b} - \mathbf{a}$ : end point minus start point. Doing it the other way round reverses the vector and every sign with it.
A position vector starts at the origin: the point $P(3,-1,2)$ has position vector with those components — no subtraction needed.
Keep columns aligned: subtract component by component, and write them stacked. Most slips here are arithmetic, not method.
Distinguish points from vectors: answer with coordinates when asked for a point, and with a column or $i, j, k$ form when asked for a vector.

Example 1

P is the point $(5, -6, 9)$ and Q is the point $(3, 2, 5)$ .

1. Find $\vec{PQ}$ :

\begin{aligned} \vec{PQ} &= \vec{q} - \vec{p} \\ &= \begin{pmatrix} 3 \\ 2 \\ 5 \end{pmatrix} - \begin{pmatrix} 5 \\ -6 \\ 9 \end{pmatrix} = \begin{pmatrix} -2 \\ 8 \\ -4 \end{pmatrix} \end{aligned}

2. Find the distance between P and Q (magnitude of $\vec{PQ}$ ):

\begin{aligned} |\vec{PQ}| &= \sqrt{(-2)^2 + 8^2 + (-4)^2} \\ &= \sqrt{4 + 64 + 16} \\ &= \sqrt{84} \\ &= \sqrt{4 \times 21} = 2\sqrt{21} \end{aligned}