Vectors (Further Pure 1)

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: $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$

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 $\mathbf{b}$.

Forces like wind and water are vectors, with both size and direction

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

Write from a cross-product magnitude of 30

• Two vectors have magnitudes 5 and 6, and the angle between them is 90 degrees.
• Find the magnitude of their vector product. Swapping the order reverses the product.