5-7 Vectors Flashcards
Vector Equation of a line in 3D space
r = [x0, y0, z0], + t(a,b,c)
Parametric equations of lines in 3D space
x = x0 + ta
y = y0 + tb
z = z0 + zc
symmetric equations of lines in 3D space
(x - x0)/a = t
(y - y0)/b = t
(z - z0)/c = t
given the normal and direction vectors, how can you tell if two lines are parallel?
n*d = 0
how to find normal vector, given a point?
Flip (x, y) and switch one sign.
Equation for DISTANCE of a point (x, y) to a line (Ax + By + C)
|Ax1 + By1 + C|
_________________
√A^2 + B^2
How to know if two lines are parallel?
Direction vectors are scalar multiples.
What are the 3 possible ways lines can intersect? How many points of intersection exist in each?
1: parallel lines, no intersection.
2: intersecting lines, 1 point of intersection.
3: coincident lines, infinite points of intersection.
You are given two lines in vector form. List the steps you would take to determine their relationship.
1 - check direction vectors. Scalar multiples?
2 - change from vector form to parametric (L1 = t, L2 = s)
3 - make parametric equal. para x, L1 = para x L2. Solve.
4 - pick 2 equations where either t or s cancel out when added together. Find one variable, substitute and find the other.
5 - take an equation from step 3, LS = RS check. If equal, point has been found.
6 - put t/s values back into parametric for either L1/L2. Find point.
What are the four line relationships that can occur in 3D space?
1 - parallel and distinct
2 - parallel and coincident
3 - not parallel, 1 point of intersection
4 - not parallel, skew
vector equation of a plane
r (x,y,z) = (x0, y0, z0) + s(a,b,c) + t(a,b,c)
Parametric equation of a plane (ex. x)
x = x0 + a1s + a2t
you are given a line and a plane. Describe how you would find how they intersect.
Place parametric line equations into the plane. If 0t = 0, line is in plane. If 0t = #, the line and plane are parallel. If t= #, place t back into parametric to find POI.
You are given two planes. Describe how you would find it and how they intersect.
1 - compare normals. Scalar multiples? If yes, parallel. Check for points.
2 - subtract planes from each other. Set a value to t, look for LINE of intersection.
You are given THREE planes. Describe how you would check for intersections.
1 - check normals of planes. If scalar multiples, check points.
2 - optional: check if coplanar (d1 x d2) * d3
3 - subtract planes 1/2 and 1/3. Check both answers. If same equation (or multiples) there is a line of intersection. If not, there is a point.
4 - set values to t.
