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
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|=
Normal vector to plane Ax+By+CZ=D
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
