Vectors

Formulae/methods for first year university vectors. No matrices.

dot product calculation

in $\mathbb{R}²$, (a₁b₁ + a₂b₂)

where u = (a₁, a₂), v = (b₁, b₂)

What is a level curve?

Any (multivariable) function for which

$f\left(x,y\right)=c$ where c is a constant.

cross product calculation in $\mathbb{R}^3$

Calculate determinant of the matrix, for x,y,z.

What property does the cross product have?

The cross product is orthogonal to both/all vectors that are crossed.

i.e. a x b is orthogonal to a and b

What does it mean for vectors to be orthogonal? What about a vector orthogonal to a plane?

Vectors are perpendicular to each other.

Vector is perpendicular to the plane.

Formula for area of a triangle?

$$A=\frac12||a\times b||$$

What is the scalar triple product (formula)?

$$u\cdot\left(v\times w\right)$$

How do I calculate the volume of a parallelepiped

Magnitude of triple scalar product

$$\left|u\cdot\left(v\times w\right)\left|\right.\right.$$

What is a parallelepiped?

A 3D parallelogram

unit vector for i, j, k

$$\bm{\hat{i}}=\left(1,0,0\right),\bm{\hat{j}}=\left(0,1,0\right),\bm{\hat{k}}=\left(0,0,1\right)$$

How to determine if two vectors are orthogonal

If the dot product of two vectors is 0, the vectors are orthogonal to each other.

What is the implicit equation of a line? (also known as cartesian here)

ax + by + c = 0

How can we find the implicit equation of a line from vectors?

For v = (v1, v2)

v^x = (-v2, v1) (this is orthogonal to v)

$$\left(\mathbf{r}-\mathbf{p}\right)\cdot\mathbf{v}^{\char"03A7 }=0$$

What is the parametric equation of a line

$\mathbf{r} = \mathbf{p} + \lambda \mathbf{v}$ where p is the position vector of a point P on the line and v is a direction vector.

What is the equation of a plane?

A plane has a parametric equation (found the same way as for a line). A plane has a vector orthogonal to it, n.

The implicit equation of the plane is given by

$$\left(\mathbf{r}-\mathbf{p}\right)\cdot\mathbf{n}=0$$

We can set n as u x v, where u and v are both parallel to the plane.

Properties of the gradient ($\nabla$)

• perpendicular to level curve

• $|\nabla f\left|\right.$ is equal to the maximum rate of directional change

• -$|\nabla f\left|\right.$ is equal to the minimum rate of directional change