Chapter 3 (Vectors) Flashcards
Length of a vector (norm)
sqrt[(v1^2) + (v2^2) + … + (vn^2)]
What is a unit vector and how do you find the direction of a vector?
- Unit vector = vector of length 1
- Direction of a vector = [1/ ||v||] x v
–> unit vector in direction of vector
U (dot) V = ?
U (dot) V = ||U|| ||V|| cos (theta)
proj aU (projection of U on a) = ?
pro aU = [(U (dot) a)/ ||a||2)] x a
vector component of U orthogonal to a
U - proj aU
norm of proj aU = ?
[|U (dot) a|]/||a||2
distance between a point and a line
D = |axo + byo + czo + d|/sqrt[(a^2) + (b^2) + (c^2)]
Steps for plane passing through 3 points
- cross product of both vectors starting from one initial point
–> to find n - Point normal form
Steps for plane through one point and perpendicular to 2 planes
- cross product of n1 and n2 (of the two other planes)
–> to find n - Point normal form
Steps for plane through 2 point and perpendicular to one plane
- cross product of (L –> AB) and (P2) (since v // n2)
–> to find n - Point normal form
Vector equation of a plane
(p) = po + v1t1 + v2t2
–> v1 and v2 = noncollinear vectors
Vector equation of a line
p = po + vt
Parametric equation of a line
x = xo +ta
y = yo + tb
z = zo + tc
Intersection of (L) and (P)
- Entire line –> (L) is in (P)
–> (L) // (P) <–> v perp n
–> v (dot) n = 0 - Empty set –> (L) // (P) but outside of (P)
–> (L) // (P) <–> v perp n
–> v (dot) n =0 - Point –> (L) NOT // (P) and intersects (P)
–> v (dot) n NOT EQUAL to 0
Intersection of 2 (P)
- Line –> (P1) NOT // (P2)
–> n1 NOT // n2 - Plane –> (P1) // (P2) + coincides (identical!)
–> n1 // n2 - Empty set –> (P1) // (P2) BUT DON’T coincide
–> n1 // n2
Direction vector of the line of intersection of 2 (P)
v = n1 x n2 (cross product)
Intersection of 2 (L)
- Empty set –> (L1) // (L2) BUT DON’T coincide
v1 // v2
–> (L1) NOT // (L2) + DON’T coincide
(skew lines) - Line –> (L1) // (L2) and coincide (identical!)
–> v1 // v2 - Point –> (L1) NOT // (L2) BUT both in same (P)
–> lie on same plane
Distance between point to (P)
D = |axo + byo + czo + d|/sqrt[(a^2) + (b^2) + (c^2)]
Steps distance between 2 // (P)
D ((P1), (P2)) = D ((A), (P1))
- Find A = (xo, yo, zo) from (P2)
–> set y, z = 0 and solve for x - Find (a, b, c) from (P1)
–> n = (a, b, c) - D =|axo + byo + czo + d|/sqrt[(a^2) + (b^2) + (c^2)]
Steps distance between (P) and // (L)
D ((L), (P)) = D ((A), (P))
- Find A = (xo,yo,zo) from (L)
- Find (a, b, c) from (P)
–> n = (a, b, c) - D =|axo + byo + czo + d|/sqrt[(a^2) + (b^2) + (c^2)]
Steps distance between point and (L)
- Find AB from distance between point given and point on (L)
- Find v from eq of line
D = ||AB x v||/||v||
Point on (P) closest to given point
Point of intersection between (P) and (L) perp to (P) and passing through given point
Method:
1. v=n of (P)
2. Find eq of (L)
3. Plug (x,y,z) in (P) eq and solve for t and then replace in eq to get point of intersection
Point on (L) closest to given point
Point on (L) such that vector from given point to point on (L) is perp to (L)
Method:
1. Point on (L) : same eq as the (L)
2. Solve for t using dot product (with the point on (L) and v of (L))
3. Replace in eq
What are skew lines?
(L1) NOT // (L2) and DO NOT intersect
–> on different planes
Distance between skew lines
D = ||proj (v1 x v2) PQ||
= |PQ (dot) (v1 x v2)|/||v1 x v2||
–> tripe scalar product
||U x V ||= ?
||U x V|| = ||U|| ||V|| sin(theta)
Area of parallelogram
A = ||U x V||
Triple scalar product
U (dot) (V x W) = V (dot) (W x U) = W (dot) (U x V)
Volume of parallelepiped
V = U (dot) (V x W)
NOTE: if U (dot) (V x W) = 0
–> vectors don’t lie on the same plane
