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projection of vector u on v (projv(u)) |
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angle between vectors u and v |
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arccos(dot product/(abs(v)*abs(u)) |
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normal line from point and gradient (vector equation for a line) |
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vectors are perpendicular/orthogonal if... |
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scalar projection of u on v |
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the cross product (u x v) |
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(abs(u)abs(v)sin(theta))n or the determinant |
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triple scalar/box product abs(u x v).w) |
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abs(u x v)abs(w)abs(cos(theta)) area of parallelogram sided box |
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parametric equation for a line |
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through (x0,y0,z0) parallel to v=v1i+v2j+v3k x= x0 +tv1 y= y0 +tv2 z=z+tv3 |
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distance from a point to a line in space |
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abs(vectorPS x v)/(abs(v) |
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component equation for a plane in space |
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A(x-x0)+B(y-y0)+C(z-z0) =0 |
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line of intersection of planes |
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find vector normal to the 2 normal vectors (take cross product) then use vector to get parametric equation of line |
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distance from point to a plane |
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abs(vectorPS . n/abs(n)) where n is vector normal to the plane |
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same as with vectors, but use normal vectors |
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arc length differential ds= |
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rectangular to polar coordinates |
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x=rcos(theta) y=rsin(theta) r^2 = x^2 + y^2 tan(theta)=y/x P(r,theta) |
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integral from alpha to beta of sqrt(r^2 + (dr/dtheta)^2) with respect to theta |
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vector equation for projectile motion |
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(vo cos(theta)t)i+(vo sin(theta))t-1/2gt^2)j |
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v/abs(v) in the direction of motion at time t |
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k=1/radius= 1/speed * (dT/dt) |
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N= 1/k (dTds) = derivative of T/magnitude of derivative of T |
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a= derivative of velocity *T +k*magnitude of speed^2 *N |
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