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| Angle between two vectors |
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Definition
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Definition
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| Angle in a triangle given 3 points |
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Definition
draw vectors from angle origin point to auxilliary points. Apply the formula for an angle to the dot product of these two vectors.
cos^-1 (AB.AC/|AB||AC|) |
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| u and v vectors are ___________iff the dot product of u and v... |
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Definition
| ... perpendicular ... = 0 |
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Definition
| 0 is perpendicular to every vector |
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Definition
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| Scalar component of u in the direction of v |
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Definition
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Definition
F x cos x d
(|F|cos)|D|
F.D |
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Term
| Vector Equation for a Line |
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Definition
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Term
| Paremetric equation for a line and point on line |
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Definition
v = i, j, k
p = x0, y0, z0
x = x0 + i; y = y0 + j; z = z0 + k |
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| Parametrize a line given 2 points. |
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Definition
P = x0, y0, z0; Q = x, y, z.
t = 0:
x0 + x-x0; y0 + y-y0; z0 + z-z0;
0 <= t <= 1 |
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| Time, speed, initial position and unit vector given. Find destination. |
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Definition
r(t) = r0 + t(v)u
r0 = initial position. u = unit vector. t = time. v = velocity. |
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| Given a point and a line determine distance from point to line. |
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Definition
Parameterize from line and point.
x = 1 + t; y = 3 - t; z = 2t
S ( 1, 1, 5) (given)
P (1, 3, 0) (Derrived from first terms on line)
v = i, -j, 2k (derrived from second terms on line)
d = |PS x v|/|v| |
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Term
| Find equation for a plane given p0 perpendicular to n |
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Definition
Combine the two given into a component equation
A(x - x0) + B(y - y0) + C(z - z0) = 0 |
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| Find equation for a plane given 3 points. |
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Definition
Draw vectors from an arbitrary origin and take the cross product of those two vectors.
AB x AC = v
Combine origin point with v.
i(x-a1) + j(y-a2) + k(z-a3) = 0 |
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Term
Two planes are parallel iff
If they intersect then... |
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Definition
Their normals are parallel (n1 = kn2)
They are not parallel and have an intersection vector. |
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| Find the vector of two intersecting planes given 2 normal vectors for each plane. |
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Definition
| Take the cross product of the normal vectors. |
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Term
| Find parametric for line in which given two plane normal vectors intersect. |
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Definition
1) n1 x n2 = intersection vector.
2) Find a point by setting 0 for one of the x y z coords on a point and solving for the other 2 variables simultaneously. (make the plane equations equal to the same value then equate the equations to eachother)
3)Combine the new point and the vector to create the parameters.
x = x0 + v1; y = y0 + v2; z = z0 + v3 |
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| Find intersection of a line and a plane given x, y, z paramtereized and a plane normal vector with a value. |
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Definition
x0 + t, y0 + t, z0 + t lies in the plane if it's coordinates satisfy the equation of the plane.
v1(x0 + t) + v2 (y0 + t) + v3(z0 + t) = 6 (OR WHATEVER THE PLANE VALUE IS)
(discover t through this equation)
Plug t into original parameters for new point for intersection of line and plane. |
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| GIVEN: A point suspended from plane S and the plane equation, find the distance from s to the plane. |
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Definition
There is an arbitrary P such that when we draw PS vector and take the absolute value of n we may plug them into.
d = (PS. (n/|n|)) |
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Term
GIVEN 2 plane equations
FIND angle bw planes |
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Definition
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Term
| vector perpendicular to a plane GIVEN 3 points |
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Definition
| Draw 2 vectors from one point of origin, take the cross product of the vectors. |
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Term
| Finding a unit normal to a plane given 3 points |
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Definition
Take vectors from single point of origin and cross product of vectors.
(PQ X PR) / (|PQ x PR|) |
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Term
| Given a vector, determine projection and distance from base vector (vector parallel to v and vector perpendicular to v) |
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Definition
projection vU = [(u.v)/|v|^2]v
parallel = projection
perpendicular = u - projection |
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Term
| formula for volume of parallelpiped given 3 parameters. |
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Definition
Take cross and dot product of parameters.
(v x u) . w
w1(v2 x u3 + v3 x u2) - w2(v1 x u3 + u1 x v3) + w3(v1 x u2 + u1 x v2) |
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