find triple product. [a*(b x c)]

Given 3 vectors, prove they are coplanar.

find two vectors from 3 points. calculate the magnitude of the cross product of them.

|AB x AC|=Area of parallelogram

|AB x AC|=Area of parallelogram

Find area of parallelogram given 3 points

find two vectors from 3 points, calculate magnitude of their cross product. Divide by 2.

|AB x AC|/2 = Area of triangle

|AB x AC|/2 = Area of triangle

Find area of triangle given 3 points

calculate cross product

find a nonzero vector orthogonal (perp.) to 2 (given) vectors

calculate two vectors from three given points.

calculate cross product.

calculate cross product.

find a nonzero vector orthogonal (perp.) to plane through 3 (given) points

calculate cross product of two vectors.

find triple product. a*(b x c)

find magnitude of triple product |a*(b x c)|

find triple product. a*(b x c)

find magnitude of triple product |a*(b x c)|

find volume of parallelepiped with 3 (given) vectors

vector equation = given point + t*

find vector equation of line through point P and parallel to vector V.

vector equation = (1-t)*(given point) + t*(vector)

find vector equation of line SEGMENT through points P and R, and parallel to vector V.

a point and a parallel vector

a line in space is determined by:

a point and a perpendicular vector

plane in space is determined by:

perpendicular to that plane

a “NORMAL” vector to a plane is:

V(a,b,c) and P=

plane = a(x-x0)+b(y-y0)+c(z-z0)=0

plane = a(x-x0)+b(y-y0)+c(z-z0)=0

find an equation of plane with given point and normal vector

find two vectors with three points

calculate cross product to find perp. vector

use one point and calculated vector in equation

calculate cross product to find perp. vector

use one point and calculated vector in equation

find equation of plane that passes through 3 (given) points

plug parametric equations into plane equation.

find value (t)

plug value (t) back into parametric equations.

find value (t)

plug value (t) back into parametric equations.

find point where given parametric equations and given plane equation intersect

use normal planes to calculate dot product

use dot product equation to find angle`

use dot product equation to find angle`

find angle between two planes (given equations)

plug 0 in for one variable in both plane equations.

find the other two point values –> finds point.

calculate cross product of normal vectors of planes.

use vector and point for line equation/symmetrics

find the other two point values –> finds point.

calculate cross product of normal vectors of planes.

use vector and point for line equation/symmetrics

find symmetic equations for line of intersection of two given planes

determine normal vectors.

plug in 0 for two values of one plane equation to find a point.

use distance equation with calculated point and other plane.

plug in 0 for two values of one plane equation to find a point.

use distance equation with calculated point and other plane.

find distance between two planes (given plane equations)

find derivative of x, y and z in vector then square each, add together and take square root of entire thing. calculate for point b, subtract the value calculated for point a

arc length of r(t)

calculate the partial derivative of each variable, put into vector form

find the gradient vector

find the gradient vector, use vector (given) find magnitude, then find unit vector.

multiply the unit vector by the gradient

multiply the unit vector by the gradient

find the directional derivative

use vector equation to find equation for the line. then find the given parametric equations from the line

find the parametric equation given two points that form a line segment

__^2+__^2=___ where the variable that is left out is the axis around which it is centered

cylinder equation

A vector has three components , a scalar is just one number or one component (x)

whats the difference between a vector and a scalar?

add their components separately

<1,2,3>+<4,5,6>=<5,7,9>

<1,2,3>+<4,5,6>=<5,7,9>

how do you add two vectors algebraically

if their dot product = 0

how do you determine if two vectors are orthogonal

if their cross product = 0

how do you determine if two vectors are parallel

find 2 vectors from 3 points. Find the magnitudes. A*B=|A||B|cos(theta) and AxB=|A||B|sin(theta)

given 3 points, find cosine/sine of an angle between 2 vectors

cross product of 2 vectors. divide by 2

find area of a triangle given 3 vertices

point=(1,2,3) cross product of 2 vectors=

put into equation, A(x-1)+B(y-2)+C(z-3)=0

put into equation, A(x-1)+B(y-2)+C(z-3)=0

find equation of plane of a point through 3 given points

use distance equation on formula sheet

find distance from origin to plane, through three given points.

v(t)=r'(t) so compute derivative of r(t). speed=|v(t)|

compute velocity/speed of particle, given curve r(t)

plug curve r(t) into surface equation set =0. solve for t. plug t back into r(t) again

compute at what point a particle meets a given surface

Find gradient of surface. plug point into gradient. use gradient as and point (d,e,f) and plug into A(x-d)+B(y-e)+C(z-f)=0

find equation of tangent plane to given surface at a given point

find df/dx and df/dy equations. plug point into both equations to get point (a,b,c). plug into plane equation

Find tangent plane equation to surface z=f(x,y) at given point (A,B,C)

a tangent line equation is a linear approximation. plug point into equation.

use linear approximation to find estimate of F at a point

Find magnitude of vector u. Find unit vector (u/|u|).

Duf(x,y)=(gradient)*(u/|u|)=scalar

Duf(x,y)=(gradient)*(u/|u|)=scalar

find the directional derivative of F at a point P in direction of vector u

find df/dx and df/dy. set equal to 0 to find critical points. use 2nd derivative test; D=FxxFyy-[Fxy]^2

find critical points, then classify as max, min, or saddle

(r,theta,z)

x=rcos(theta)

y=rsin(theta)

z=z

x=rcos(theta)

y=rsin(theta)

z=z

cylindrical coordinates

(rho,theta,phi)

x=rho*sin(phi)*cos(theta)

y=rho*sin(phi)*sin(theta)

z=rho*cos(phi)

x=rho*sin(phi)*cos(theta)

y=rho*sin(phi)*sin(theta)

z=rho*cos(phi)

spherical coordinates

find r(0) and r(1) points. plug into the gradient of F. Find the difference.

find the line integral (F*dr) given a function & parametric curve 0

find F(r(t)). Find r'(t).

=integral(F(r(t))*r'(t)dt

=integral(F(r(t))*r'(t)dt

evaluate line integral (F*dr) given a parametrization

ax2+by2+cz^2=1

What is the equation of an ellipsoid?

x2+y2+z2=1

What is the equation of a sphere?

x2/a+y2/b-z/c=0

What is the equation of an elliptical paraboloid?

x2+y2-z2=0

What is the equation of a cone?

x2+y2=1

What is the equation of an ellipse?

r(t)=(1-t)(initial point)+t(final point)

0

0

How to parametrize a line segment given two points

put x(t) and y(t) into F(x,y). find derivatives of each parametric equation.

=integral (F(r(t))*sqrt(dx/dt)^2+(dy/dt)^2)

=integral (F(r(t))*sqrt(dx/dt)^2+(dy/dt)^2)

compute a line integral of a function F, given parametrizations x(t) and y(t)

find gradient f(x,y,z) and plug bounds into r(t) to find points to plug into f(x,y,z). find f(b)-f(a)

evaluate F*dr given F(x,y,z) and r(t)

compute ru and rv. compute ru x rv

determine the values of u and v by setting individual components of r(u,v) to the coordinates of the given point.

plug u and v into vector (ru x rv) which is which gives (a,b,c). plug into plane equation

determine the values of u and v by setting individual components of r(u,v) to the coordinates of the given point.

plug u and v into vector (ru x rv) which is which gives (a,b,c). plug into plane equation

find tangent plane to a given parametric surface r(u,v) at given point (x,y,z)

gradient of F

find unit normal vector given z=g(x,y)

ru x rv

find unit normal given r(u,v)

scaler triple product |aº(b x c )|

find out if three points are coplaner?

|a||b|cos(theta)

|a*b|=

|a||b|sin(theta)

|a x b|=

x^2-y^2-z=0

Equation of hyperbolic paraboloid

x^2+y^2-z^2=1

Equation of hyperboloid of 1 sheet

-x^2-y^2+z^2=1

Equation of hyperboloid of 2 sheets