Vector geometry
| English | Chinese | Pinyin |
|---|---|---|
| collinear | 共线 | gòng xiàn |
| position vector | 位置向量 | wèi zhì xiàng liàng |
| parallel | 平行 | píng xíng |
| midpoint | 中点 | zhōng diǎn |
Proving points line up
- How do you prove that three points are collinear 共线 (lie on the same line)?
- Show that the vector from the first to the second is a scalar multiple of the vector from the second to the third.
Position vectors 位置向量 and $\overrightarrow{AB}$
- The position vector of a point is the vector from the origin $O$ to it.
- The vector from $A$ to $B$:
If $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4 \\ 7 \end{pmatrix}$, then $\overrightarrow{AB} = \begin{pmatrix} 4-1 \\ 7-3 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$.
Vector geometry
resultant = a + b
Combine vectors to reach a point — the resultant is the direct route.
If a and b are the position vectors of A and B, then the vector AB is:
AB = (position of B) − (position of A) = b − a.
A has position vector (1, 3) and B has position vector (4, 7). The top component of AB is?
AB = b − a: top = 4 − 1 = 3.
Parallel 平行 vectors and collinear points
- Two vectors are parallel if one is a scalar multiple of the other: $\mathbf{p} = k\mathbf{q}$.
- Three points $A, B, C$ are collinear if $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are parallel and share the point $B$.
Parallel alone isn't enough. Two vectors can be parallel without the points being collinear — they could be on separate parallel lines. You need a shared point too.
Two vectors are parallel if one is a scalar multiple of the other.
Parallel vectors point the same (or opposite) way, so one is k times the other.
To prove three points are collinear, show the vectors between them are ______ and share a point.
Collinear points have parallel vectors between them AND a shared point.
Midpoint 中点 of a segment
- $M$ is the midpoint of $AB$, where $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$.
- The midpoint's position vector is the average of the two endpoints.
The magnitude of a column vector is found with Pythagoras
M is the midpoint of AB. The position vector OM is:
OM = a + ½(b − a) = ½(a + b).
A = (2, 4) and B = (8, 10). The midpoint M has x-coordinate?
½(2 + 8) = 5.
Why vectors matter
- Video games, physics engines, and GPS all use vector mathematics to track positions and movements.
Position vectors locate points on the grid — the foundation for proving collinearity and finding midpoints.
You've got it
- $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ (end minus start)
- parallel vectors are scalar multiples; that's how you show points are collinear
- midpoint of $AB$: $\;\overrightarrow{OM} = \dfrac{1}{2}(\mathbf{a} + \mathbf{b})$