Vectors (Further Pure 1)
| English | Chinese | Pinyin |
|---|---|---|
| plane | 平面 | píng miàn |
| skew | 异面 | yì miàn |
| vector product | 向量积 | xiàng liàng jī |
| normal vector | 法向量 | fǎ xiàng liàng |
| perpendicular | 垂直 | chuí zhí |
| parallelogram | 平行四边形 | píng xíng sì biān xíng |
| anti-commutative | 反交换 | fǎn jiāo huàn |
The mathematics of 3D space
- How do you find the distance from a point to a plane 平面? The angle between two surfaces? The shortest path between two skew 异面 lines?
- Planes and the vector product 向量积 give you the tools to solve all these problems — essential for engineering, architecture, and computer graphics.
Equations of planes
- A plane can be written in several forms:
The normal vector 法向量 n is perpendicular 垂直 to the plane; the scalar-product form is n·r = d.
- Cartesian: $ax + by + cz = d$
- Vector: $\mathbf{r}\cdot\mathbf{n} = p$ (where $\mathbf{n}$ is the normal vector)
- Parametric: $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$
Worked example. A plane has normal $\mathbf{n} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and passes through $(1, 0, 2)$. Equation: $2x - y + 3z = 2(1) - 0 + 3(2) = 8$.
Vectors
a · b = |a||b|cos θ
The dot product is zero exactly when two vectors are perpendicular.
Scalar and vector products can find the shortest distance between two skew lines.
These products let you compute distances, angles and intersections in 3D, including skew-line distances.
A plane has equation 2x − y + 3z = 8. The normal vector has components (2, −1, ?). Find the missing component.
The coefficients of x, y, z in ax + by + cz = d give the normal vector (a, b, c) = (2, −1, 3).
The vector (cross) product
- The vector product $\mathbf{a}\times\mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin\theta\,\hat{\mathbf{n}}$ gives a vector perpendicular to both.
- Its length equals the area of the parallelogram 平行四边形 spanned by $\mathbf{a}$ and $\mathbf{b}$.
Order matters. $\mathbf{a}\times\mathbf{b} = -\mathbf{b}\times\mathbf{a}$. The vector product is anti-commutative 反交换 — swapping the order reverses the direction.
Forces like wind and water are vectors, with both size and direction
The vector product a × b gives a vector that is:
a × b is perpendicular to both vectors; its length is |a||b|sin θ.
The magnitude |a × b| equals the:
|a × b| = |a||b|sin θ, which is the parallelogram's area.
The vector product is commutative: a × b = b × a.
The vector product is anti-commutative: a × b = −(b × a). Swapping the order reverses the direction.
If |a| = 3, |b| = 4, and the angle between them is 30°, |a × b| = |a||b|sin θ. Find it.
|a × b| = 3 × 4 × sin 30° = 12 × 0.5 = 6.
Applications
- With scalar and vector products: distances, angles, where lines and planes meet, and the shortest distance between skew lines.
- Distance from point to plane: $d = \dfrac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}$.
The vector product is perpendicular to both vectors; its length is the parallelogram area
Finding the normal
- The normal to a plane containing two vectors $\mathbf{b}$ and $\mathbf{c}$ is $\mathbf{n} = \mathbf{b} \times \mathbf{c}$.
- This is the quickest way to find the Cartesian equation from a parametric form.
You've got it
- a plane: $ax + by + cz = d$ or $\mathbf{r}\cdot\mathbf{n} = p$
- the vector product $\mathbf{a}\times\mathbf{b}$ is perpendicular to both; $|\mathbf{a}\times\mathbf{b}|$ = parallelogram area
- use it for distances, angles, intersections, and skew-line distances