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Calculus Volume 3

Key Equations

Calculus Volume 3Key Equations

Key Equations

Distance between two points in space: d=(x2x1)2+(y2y1)2+(z2z1)2d=(x2x1)2+(y2y1)2+(z2z1)2
Sphere with center (a,b,c)(a,b,c) and radius r: (xa)2+(yb)2+(zc)2=r2(xa)2+(yb)2+(zc)2=r2
Dot product of u and v u·v=u1v1+u2v2+u3v3=uvcosθu·v=u1v1+u2v2+u3v3=uvcosθ
Cosine of the angle formed by uu and vv cosθ=u·vuvcosθ=u·vuv
Vector projection of vv onto uu projuv=u·vu2uprojuv=u·vu2u
Scalar projection of vv onto uu compuv=u·vucompuv=u·vu
Work done by a force F to move an object through displacement vector PQPQ W=F·PQ=FPQcosθW=F·PQ=FPQcosθ
The cross product of two vectors in terms of the unit vectors u×v=(u2v3u3v2)i(u1v3u3v1)j+(u1v2u2v1)ku×v=(u2v3u3v2)i(u1v3u3v1)j+(u1v2u2v1)k
Vector Equation of a Line r=r0+tvr=r0+tv
Parametric Equations of a Line xx0a=yy0b=zz0cxx0a=yy0b=zz0c
Vector Equation of a Plane n·PQ=0n·PQ=0
Scalar Equation of a Plane a(xx0)+b(yy0)+c(zz0)=0a(xx0)+b(yy0)+c(zz0)=0
Distance between a Plane and a Point d=projnQP=|compnQP|=|QP·n|nd=projnQP=|compnQP|=|QP·n|n
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