Vectors

Vectors

1. Forms

   • Column
     • (X,Y,Z)
   • Component
     • Xi+Yk+Zk

2. Scalar(Dot) Product

   • a●b=|a||b|cosθ
   • cosθ = (a●b)/(|a||b|)
   • a●b=x1x2+y1y2+z1z2
   • If a●b=0 then ab is perpendicular

3. Vector(Cross) Product

   • |axb| = |a||b|sinθ
   • Area of a parallelogram
     • Magnitude of the adjacent sides a,b axb=c area=|c|
     • Scalar Triple Product: volume of parallelepiped= |u●(wxv)|
       • If volume=0, the 3 vectors are coplanar

4. Equation of a line

   • Point(X,Y,Z) Direction vector v= (a,b,c)
     • Vector form: r=r。+tv
     • Cartesian form: {(x-x1)/a}= {(y-y1)/b}= [(z-z1)/c]
     • Parametric form: {x=x1+at {y=y1+bt {z=z1+ct
   • Point-Line M1:d=|BAxv|/|v| Where B is a point on the line and v is the direction vector
     • M2: Find the closest point on the line (X.Y,Z)=(X1,Y1,Z1)+t(v1,v2,v3) to the given point A(X1,Y1,Z1)