Calculus Volume 3

Key Equations

Calculus Volume 3Key Equations

Key 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 $uu$ and $vv$ $cosθ=u·v‖u‖‖v‖cosθ=u·v‖u‖‖v‖$ Vector projection of $vv$ onto $uu$ $projuv=u·v‖u‖2uprojuv=u·v‖u‖2u$ Scalar projection of $vv$ onto $uu$ $compuv=u·v‖u‖compuv=u·v‖u‖$ 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 $x−x0a=y−y0b=z−z0cx−x0a=y−y0b=z−z0c$ 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 $d=‖projnQP→‖=|compnQP→|=|QP→·n|‖n‖d=‖projnQP→‖=|compnQP→|=|QP→·n|‖n‖$
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