Calculus 3 Vectors and geometry of space
Find an equation of a sphere with radius r and center C(h, k, l).
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
Unit vector
A vector whose length is 1
what quadric surface does this equation represent z = y^2 - x^
Hyperbolic paraboloid
Area of a triangle with vertices P(1, 4, 6) Q(-2, 5, -1), R(1, -1, 1)
PQ cross PR is equal to <-40, -15, 15>. the area of the parallelogram is equal to the length of the cross product, which is 5 sqrt(82)
dot product of a * b
a * b = a1b1+a2b2+a3b3
using scalar triple product: a<1, 4, -7>, b<2, -1, 4>, c<0, -9, 18> they are also coplanar
a dot b cross c is equal to 0, the scalar triple product indicates the points lie within the same plane
Scalar Triple Product
a dot b crossed c.
Normal vector of plane
a plane in space is determined by a point p-not(x-not, y-not, z-not) in the plane and a vector n that is orthogonal to the plane. n is orthogonal to every vector in the plane, so an arbitrary point may be used.
Scalar Equation of a Plane
a scalar equation of the plane through the point p-not(x-not, y-not, z-not) with normal vector n equals <a, b, c>
angle theta between vector a and b
a times b equals absolute power of a times absolute power b time cos of theta
vector equation of the plane
a(x - x-not) + b(y - y-not) + c(z - z-not) = 0
angle between vector a and b is equal to
absolute power of a cross b is also equal to absolute power of a times absolute power of b sin theta
find the angle between the vectors a = <2, 2, -1> and b = <5, -3, 2>
absolute power of a would be equal to 3, the absolute power of b would be equal to sqrt(38) the dot product of a and b would equal 2. the answer would be arccosine 2 over 3 sqrt(38)
corollary angle of theta between the nonzero vectors a and b
cos theta is equal to the dot product of a and b over the absolute power of a times absolute power of b
linear equation
d = -(ax-not + by-not + cz-not)
what quadric surface does this equation represent x^2 + y^2/9 + z^2/4 = 1
ellipsoid
what quadric surface does this equation represent z = 4x^2 + y^2
elliptic paraboloid
distance equation between a point and plane
equaling n=<a,b,c> b=<x1-x, y1-y, z1-z>
identify the method to find an equation of the plane passing through the points P(1, 3, 2), Q(3, -1, 6) and R(5, 2, 0)
find the vectors corresponding to PQ and PR. since both a and b lie in the plane, their cross product a cross b is orthogonal to the plane and can be taken as the normal vector. n=aXb which then equals 12i + 20j + 14k - the normal vector (n). Using point P and the normal vector(n) the equation of the plane is 12(x - 1) + 20(y - 3) + 14(z - 2) = 0 simplifying to 6x + 10y +7z = 50
what quadric surface does this equation represent x^2/4 + y^2 - z^2/4 = 1
hyperboloid of one sheet
skew line test of direction vectors <1, 3, -1> and <2, 1, 4>
not parallel due to the skew line test
parametric equation
parametric equations for a line passing through the point x-not y-not z-not and parallel to the direction vector a, b, c are x equals x-not + a t, y equals y-not + b t, z equals z-not + c t
find an equation of the plane through the point (2, 4, -1) and the normal vector n = <2, 3, 4>
putting a = 2 b = 3 c = 4, x-not = 2 y-not = 4 z-not = -1 we get 2(x-2) +3(y-4) + 4(z+1) = 0 simplifying to 2x + 3y + 4z = 12
vector equation
r is equal to r-not + t v.
vector equation of a line segment from r-not to r1
r of t equals (1-t)r-not +t r1
expanded vector equation
r represents <x, y, z,> equal to r-not which is <x-not, y-not, z-not> plus <ta, tb, tc>
Turn a parametric into a symmetric equation
solve for t
Symmetric equation of L
t equals x - x-not over a equals y - y-not over b equals z - z-not over c.
direction angle of a vector
the direction angles of a nonzero vector a are the angles alpha, beta and gamma. which equal cos alpha equals i over absolute power of a, cos beta is equal to j over the absolute power of a. cos gamma is equal to k over the absolute power of a.
Area of a parallelogram
the length of the cross product a cross b is equal to the area of the parallelogram determined by a and b
vector of a plane that passes through the points P(1, 4, 6) Q(-2, 5, -1) and R(1, -1, 1)
the vectors PQ cross PR are perpendicular to both PQ and PR and is therefore perpendicular to the plane through P, Q, R. PQ is equal to -3j + j - 7k and PR is equal to -5j -5k. the cross product of PQ PR is equal to -40i - 15j + 15k
Volume of a parallelepiped
the volume of a parallelepiped is determined by the vectors a, b, c and is the magnitude of their scalar triple product. which is the absolute power of a dot b cross c
nonzero vectors corollary
two nonzero vectors a and b are parallel if and only if a cross b equals 0
orthogonal vectors
two vectors are orthogonal if and only if the dot product of a and b are equal to 0
Equal vectors
vectors are equal if and only if their corresponding components are equal.
Describe a line segment given the parametric equation x=2+t y=4-5t z=-3+4t
we first convert the parametric equation into a vector equation r of t equals 2 + t, 4 - 5t, -3 + 4t 0 < t < 1.
symmetric equation for line L through two given points
x - x-not over x1 - x-not equals y - y-not over y1 - y-not equals z - z-not over z1 - z-not. P-not equates to all -not values, P1 equates to x1, y1, z1.
at point 5, 1, 3 choose a parallel vector 2i + 8j -4k
x equals 5 + 2t y equals 1 + 8t and z equals 3 - 4t.
The Three scalar equations
x equals x-not + a t, y equals y-not + b t, z equals z-not + c t
space curves
x= f(t) y= g(t) z= h(t)
Find the unit vector for 2i - j - 2k
|2i - j - 2k| = sqrt(2^2+-1^2+2^2) = sqrt(9) = 3
