Vectors (Further Pure 1)
Planes and the vector product
- A plane can be written $ax + by + cz = d$, or $\mathbf{r}\cdot\mathbf{n} = p$, or $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}$.
- 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 spanned by $\mathbf{a}$ and $\mathbf{b}$.
Practice
The vector product a × b gives a vector that is:
a × b is perpendicular to both vectors; its length is |a||b|sin θ.
Practice
The magnitude |a × b| equals the:
|a × b| = |a||b|sin θ, which is the parallelogram's area.
What you can find
- With scalar and vector products: distances, angles, where lines and planes meet, and the shortest distance between skew lines.
Practice
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.
You've got it
Key idea
- 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