Vectors Formula Sheet

Vectors Formula Sheet

Component Form & Magnitude

A=(x_1, y₁) B=(x₂, y₂)

• Component Form:  $<x_2-x_1,\ y_2-y_1>\ =\ <v_1,\ v_2>$  

• Magnitude:  $\parallel v\parallel=\sqrt{\left(v_1\right)^2+\left(v_2\right)^2}$  

Unit Vector

• u = $<\frac{v_1}{\parallel v\parallel},\ \frac{v_2}{\parallel v\parallel}>$  

  
  

•  $\parallel v\parallel=\sqrt{\left(v_1\right)^2+\left(v_2\right)^2}$  

Direction Angle

•  $\theta=\tan^{-1}\left(\frac{b}{a}\right)$  

•  $a=v_1,\ b=v_2$  

Dot Product & Is it Orthogonal

u=<u₁, u₂> , v=<v₁, v₂>

• Dot Product: $u\cdot v=\left(u_1\right)\left(v_1\right)+\left(u_2\right)\left(v_2\right)$  

  
  

•  $If\ u\cdot v=0,\ it's\ orthogonal$  

Angle Between Two Vectors

•  $\theta=\cos^{-1}\left(\frac{u\cdot v}{\parallel u\parallel\times\parallel v\parallel}\right)$  

• Step 1:  $u\cdot v$  (numerator)

• Stept 2:  $\parallel u\parallel$  

• Step 3:  $\parallel v\parallel$  

• Step 4:  $\parallel u\parallel\times\parallel v\parallel$  (denominator)

• Step 5: Plug the answers from steps 1 & 4 into the formula

Vector Projections (Find w₁ & w₂)

•  $w_1=<\frac{u\cdot v}{\parallel v\parallel^2}>\left(v\right)$  

• Step 1:  $u\cdot v$  

• Step 2:  $\parallel v\parallel^2$  

• Step 3: Plug into formula & distribute

• Step 4:  $w_2=u-w_1$  

Cross Product & Area of Parallelogram

u = <u₁, u₂, u₃> , v = <v₁, v₂, v₃>

• u x v:

• Step 1:  $\left(u_2\right)\left(v_3\right)-\left(u_3\right)\left(v_2\right)$  

• Step 2:  $\left(u_1\right)\left(v_3\right)-\left(u_3\right)\left(v_1\right)\ change\ the\ sign$  

• Step 3:  $\left(u_1\right)\left(v_2\right)-\left(u_2\right)\left(v_1\right)$  

• Area =  $\parallel\sqrt{u\times v}\parallel$  

Volume of a Parallelepiped

t=<t_1, t₂, t₃>, u=<u₁, u₂, u₃>, v=<v₁, v₂, v₃>

• volume = $t\cdot\left(u\times v\right)$