Products

Dot Product

Definition

Given 2 nonzero vectors U\vec{U} and V\vec{V} in 2D or 3D, their dot product is UV=UVcosθ\vec{U}\cdot\vec{V}= |\vec{U}||\vec{V}|\cos{\theta} where θ\theta is the angle between U\vec{U} and V\vec{V} with 0θπ0\leq\theta\leq\pi

Equation

U=U1,U2,U3\vec{U}=\langle U_1,U_2,U_3\rangle

V=V1,V2,V3\vec{V}=\langle V_1,V_2,V_3\rangle

UV=U1V1+U2V2+U3V3=UVcosθU \cdot V = U_1V_1+U_2V_2+U_3V_3=|\vec{U}||\vec{V}|\cos{\theta}

Key Notes

  • Dot product is commutative

Orthogonal Vectors

Two vectors are orthogonal if UV=0\vec{U}\cdot\vec{V}=0. In 2D and 3D spaces, orthogonal means perpendicular (π2\frac{\pi}{2} angle between them)

Key Notes

  • There are an infinite number of orthogonal vectors to any one vector when length is not specified

Projections

The orthogonal projection is a vector that goes along U\vec{U} with length Vcosθ|\vec{V}|\cos{\theta}

Equation

projUV=UUVcosθproj_{\vec{U}}\vec{V}= \frac{\vec{U}}{|\vec{U}|}|\vec{V}|\cos{\theta}

Cross Product

Definition

Given two nonzero vectors uu and vv in I ⁣R3{\rm I\!R}^3, the cross product is defined as u×v=uusinθn\vec{u} \times \vec{v} = |\vec{u}||\vec{u}|\sin{\theta}\vec{n} where 0θπ0\leq\theta\leq\pi is the angle between u\vec{u} and v\vec{v} and n\vec{n} is the unit vector which is normal (perpendicular) to u\vec{u} and v\vec{v} whose direction is determined by the right hand rule

Equation u×v=uusinθn\vec{u} \times \vec{v} = |\vec{u}||\vec{u}|\sin{\theta}\vec{n}

Key Notes

  • u\vec{u} and v\vec{v} are parallel if and only if u×v=0\vec{u}\times\vec{v}=0
  • If u\vec{u} and v\vec{v} are two sides of a parallelogram, then the area of the paralllelogram is u×v=uvsinθ|\vec{u}\times\vec{v}| = |\vec{u}||\vec{v}|\sin{\theta}
  • v×u=(u×v)\vec{v}\times\vec{u}=-(\vec{u}\times\vec{v})
  • u×v\vec{u}\times\vec{v} is orthogonal to u\vec{u} and v\vec{v} which means that (u×v)u=0(\vec{u}\times\vec{v})\cdot\vec{u} = 0 and (u×v)v=0(\vec{u}\times\vec{v})\cdot\vec{v} = 0

Cross Product Distribution

With respect to addition

  • u×(v+w)=u×v+u×w\vec{u}\times(\vec{v}+\vec{w}) = \vec{u}\times\vec{v} + \vec{u}\times\vec{w}

  • (u+v)×w=u×w+v×w(\vec{u}+\vec{v})\times\vec{w} = \vec{u}\times\vec{w} + \vec{v}\times\vec{w}

If c is a scalar

-(cu×v=c(u×v)=u×(cv)(c\vec{u}\times\vec{v}=c(\vec{u}\times\vec{v})=u\times(c\vec{v})

Cross Products of Unit Vectors

flowchart LR i --> j --> k k --> i
  • iˆ×jˆ=kˆ\^{i}\times\^{j}=\^{k}
  • jˆ×kˆ=iˆ\^{j}\times\^{k}=\^{i}
  • kˆ×iˆ=jˆ\^{k}\times\^{i}=\^{j}
  • jˆ×iˆ=kˆ\^{j}\times\^{i}=-\^{k}
  • kˆ×jˆ=iˆ\^{k}\times\^{j}=-\^{i}
  • iˆ×kˆ=jˆ\^{i}\times\^{k}=-\^{j}
  • iˆ×iˆ=0\^{i}\times\^{i}=0
  • jˆ×jˆ=0\^{j}\times\^{j}=0
  • kˆ×kˆ=0\^{k}\times\^{k}=0

Right Hand Rule

Definition

Finds the direction of the cross product

Example

Find the direction of a×b\vec{a} \times \vec{b}

Using your right-hand:

  1. point your index finger along a\vec{a}
  2. point your middle finger along vector a\vec{a}
  3. a×b\vec{a} \times \vec{b} will be in the direction of your thumb.