Calculus Volume 3

# Key Equations

Calculus Volume 3Key Equations
• Distance between two points in space:
$d=(x2−x1)2+(y2−y1)2+(z2−z1)2d=(x2−x1)2+(y2−y1)2+(z2−z1)2$
• Sphere with center $(a,b,c)(a,b,c)$ and radius r:
$(x−a)2+(y−b)2+(z−c)2=r2(x−a)2+(y−b)2+(z−c)2=r2$
• Dot product of u and v
$u·v=u1v1+u2v2+u3v3=‖u‖‖v‖cosθu·v=u1v1+u2v2+u3v3=‖u‖‖v‖cosθ$
• Cosine of the angle formed by u and v
$cosθ=u·v‖u‖‖v‖cosθ=u·v‖u‖‖v‖$
• Vector projection of v onto u
$projuv=u·v‖u‖2uprojuv=u·v‖u‖2u$
• Scalar projection of v onto u
$compuv=u·v‖u‖compuv=u·v‖u‖$
• Work done by a force F to move an object through displacement vector $PQ→PQ→$
$W=F·PQ→=‖F‖‖PQ→‖cosθW=F·PQ→=‖F‖‖PQ→‖cosθ$
• The cross product of two vectors in terms of the unit vectors
$u×v=(u2v3−u3v2)i−(u1v3−u3v1)j+(u1v2−u2v1)ku×v=(u2v3−u3v2)i−(u1v3−u3v1)j+(u1v2−u2v1)k$
• Vector Equation of a Line
$r=r0+tvr=r0+tv$
• Parametric Equations of a Line
$x=x0+ta,x=x0+ta,$ $y=y0+tb,y=y0+tb,$ and $z=z0+tcz=z0+tc$
• Symmetric Equations of a Line
$x−x0a=y−y0b=z−z0cx−x0a=y−y0b=z−z0c$
• Vector Equation of a Plane
$n·PQ→=0n·PQ→=0$
• Scalar Equation of a Plane
$a(x−x0)+b(y−y0)+c(z−z0)=0a(x−x0)+b(y−y0)+c(z−z0)=0$
• Distance between a Plane and a Point
$d=‖projnQP→‖=|compnQP→|=|QP→·n|‖n‖d=‖projnQP→‖=|compnQP→|=|QP→·n|‖n‖$