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CAlculus 3

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Distance B/w 2 points, what formula do you use?

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Calculus Math

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Mar 1, 2020

Cards (21)

Section 1

(21 cards)

Distance B/w 2 points, what formula do you use?

Front

Distance Formula= Sqrt[(X2-X1)^2 + (Y2-Y1)^2 +(Z1-Z2)^2

Back

What does v = ||v|| u trell you

Front

The magnitude( ||v||) of a vector and its direction (unit vector) = vector

Back

Volume of Parallelopiped

Front

Vpp = | a dot ( b x c) | You need 4 points, 3 vectors if Vpp = 0, 4th point is in the same plane as the other 3

Back

Magnitude/ length of a vector

Front

SQrt[ V1^2 + V2^2 + V3^2]

Back

Work

Front

||F|| ||d|| Cos() or Fxd Know theta

Back

Position Vector in R^3

Front

IF A(a1,a2,a3) and B(b1,b2,b3) then the position vector is given by AB= (b1-a1, b2-a2, b3-a3)

Back

Mid Point and Center

Front

(X1+X2)/2, (Y1+Y2)/2, (Z1+Z2)/2 - Mid Point (h,k,l): opposite signs than the ones they have in the equation

Back

Lines in 3D

Front

If a direction vector (a,b,c) =0 thent he line lies in a plane Lines are parallel if their direction vectors are parallel(Scalar Multiples)

Back

Vector Projection

Front

Comp w V times unit vector of W

Back

angle b/w 2 vectors

Front

Cos( theta)= (v dot w)/(||v|| ||w||)

Back

Scalar/Component Projection ( CompwV )

Front

(v dot w)/ ||w|| or ||v||Cos() ( if you dont know theta) ( if you know theta)

Back

Cross Product

Front

a x b is orthogonal to a, b If a and b are parallel than a x b = 0 || a x b|| = ||a|| ||b||Sin(0 a x b= determinate with < i j k> in first row a dot ( a x b) = 0 anti commutative : a xb = -( b x a )

Back

Area of a triangle & Paralelogram

Front

Ap= ||a|| ||v||Sin() Ap= ||a x v|| Atriangle= .5 (||a x b||)

Back

Dot product

Front

adds product of corresponding components of our vectors = scalar a dot b= ||a|| ||b|| cos() a is orthogonal to b if their dot products =0

Back

Torque

Front

torque = || r|| ||F||Sin()

Back

parametric equation in 3D

Front

X = Xo + at, Y = Yo +bt, Z= Zo+ ct Point (Xo, Yo, Zo) Direction vector= (a,b,c) If they give you 2 points you can use the distnace formula for vectors (b1-a1, b2-a2, b3-a3) to find the direction vector and then use any of the two points to fill in the parametric or symmetric equation

Back

Symmetric equation

Front

Solve for t on the parametric equation ( X-Xo/a ) = ( Y-Yo/b ) = ( Z-Zo/c )

Back

Direction Cosines

Front

Cos( alpha )= v1/||v|| , Cos(Beta) = v2/||v|| Cos(Y)= v3/||v||

Back

Parallel vectors

Front

scalar multiples

Back

Planes in 3D

Front

Need a normal vector (a,b,c) and a point (Xo,Yo.Zo) a(X-Xo) + b(Y-Yo) + c(Z-Zo) this equation turns into aX +bY + cZ = d (the constants) -If planes are parallel they have the same normal vector If they give you for points find the distnce b/w points PQ= a PR= b a x b = n this gives you your normal vector You can then use any point to fill in the equation 2 planes are parallel if their normal vectors are parallel( scalar multiples) 2planes are perpendicular if their normals are perpendiculars ( n1 x n2 = 0 ) Angle b/w planes is the angle b/w normal

Back

Equation for a sphere and Completing the square

Front

( X-H)^2 +(Y-k)^2 + (Z-l)^2 = r^2 To complete the square you have to group all your same variables together in descending order of degree. Then you have to find the scalars that are missing by halfing b and then squaring it and that will be your c. You have to add your c term to the other side of the equation for each type of variable. Doing this will give you r^2 on the right side and ( h,k,l) will be the values when you half b.

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