ch 12 calc

# ch 12 calc - product: v=mang(a*(b x c)) Equations-r =

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Vectors -(x-h)^2+(y-k)^2+(z-l)^2=r^2 -x^2+y^2+z^2=r^2 -a = <x(2)-x(1),y(2)-y(1),z(2)-z(1)> Dot Product -a*b=a(1)b(1)+a(2)b(2)+a(3)b(3) 1. a*a=magn(2)^2 2. a*b=b*a 3. a*(b+c)=a*b+a*c 4. (ca)*b=c(a*b)=a*(cb) 5. 0*a=0 -a*b=magn(a)magn(b)(cos(theta)) -two vectors a and b are orthogonal if and only if a*b=0 -scalar projection of b onto a: comp(a)b=a*b/magn(a) -vector projection of b onto a: proj(a)b=(a*b/magn(a))a/magn(a)=a^2*b/magn(a)^2 Cross Product -a x b = <a(2)b(3)-a(3)b(2),a(3)b(1)-a(1)b(3),a(1)b(2)-a(2)b(1)> -the vector a x b is orthogonal to both a and b -mang(a x b)= magn(a)magn(b)sin(theta) -two nonzero vectors a and b are parallel if and only if a x b=0 -the length of the cross product a x b is equal to the area of the parallelogram determined by a and b 1. a x b=-b x a 2. (ca)xb=c(a x b)=ax(cb) 3. a x (b+c)=a x b+a x c 4. (a+b) x c=a x c+b x c 5. a*(b x c)=(a x b)*c 6. a x (b x c)=(a*c)b-(a*b)c -the volume of the parallelepiped determined by the vectors a,b,c is the magnitude of their scalar triple

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Unformatted text preview: product: v=mang(a*(b x c)) Equations-r = r(0)+tv-x=x(0)+at-y=y(0)+bt-z=z(0)+ct-(x-x(0))/a=(y-y(0))/b=(z-z(0))/c-the line segment from r(0) to r(1) is given by the vector equation: r(t)=(1-t)r(0)+tr(1) 0&lt;(equal)t&lt;(equal)1-n*(r-r(0))=0-vector equation of plane: a(x-x(0))+b(y-y(0))+c(z-z(0))=0-D=magn(ax(1)+by(1)+cz(1)+d)/sqr(a^s+b^s+c^2) Surfaces-ellipsoid: x^2/a^2+b^2/b^2+x^2/c^2=1-cone: z^2/c^2=x^2/a^2+y^2/b^2-elliptic paraboloid: z/c=x^2/a^2+y^2/b^2-hyperboloid of one sheet: x^2/a^2+y^2/b^2-z^2/c^2=1-hyperbolic paraboloid: z/c=x^2/a^2-y^2/b^2-hyperboloid of two sheets: -x^2/a^2-y^2/b^2+z^2/c^2=1 Cylindrical and Spherical Coordinates-cylindrical coordinate system:-x=rcos(theta)-y=rsin(theta)-z=z-r^2=x^2+y^2-tan(theta)=y/x-z=z-(r,theta,z)-spherical coordinates:-x=psin(ol)cos(theta)-y=psin(ol)sin(theta)-z=pcos(ol)-p^2=x^2+y^2+z^2-(p,theta,ol)...
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## This note was uploaded on 01/13/2012 for the course MATH 1 taught by Professor Staff during the Spring '08 term at UC Davis.

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ch 12 calc - product: v=mang(a*(b x c)) Equations-r =

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