Chapter 3 (Vectors)

Question 1

Length of a vector (norm)

Answer 1

sqrt[(v1^2) + (v2^2) + … + (vn^2)]

Question 2

What is a unit vector and how do you find the direction of a vector?

Answer 2

1. Unit vector = vector of length 1
2. Direction of a vector = [1/ ||v||] x v
   –> unit vector in direction of vector

Question 3

U (dot) V = ?

Answer 3

U (dot) V = ||U|| ||V|| cos (theta)

Question 4

proj aU (projection of U on a) = ?

Answer 4

pro aU = [(U (dot) a)/ ||a||2)] x a

Question 5

vector component of U orthogonal to a

Answer 5

U - proj aU

Question 6

norm of proj aU = ?

Answer 6

[|U (dot) a|]/||a||2

Question 7

distance between a point and a line

Answer 7

D = |axo + byo + czo + d|/sqrt[(a^2) + (b^2) + (c^2)]

Question 8

Steps for plane passing through 3 points

Answer 8

1. cross product of both vectors starting from one initial point
   –> to find n
2. Point normal form

Question 9

Steps for plane through one point and perpendicular to 2 planes

Answer 9

1. cross product of n1 and n2 (of the two other planes)
   –> to find n
2. Point normal form

Question 10

Steps for plane through 2 point and perpendicular to one plane

Answer 10

1. cross product of (L –> AB) and (P2) (since v // n2)
   –> to find n
2. Point normal form

Question 11

Vector equation of a plane

Answer 11

(p) = po + v1t1 + v2t2
–> v1 and v2 = noncollinear vectors

Question 12

Vector equation of a line

Answer 12

p = po + vt

Question 13

Parametric equation of a line

Answer 13

x = xo +ta
y = yo + tb
z = zo + tc

Question 14

Intersection of (L) and (P)

Answer 14

1. Entire line –> (L) is in (P)
   –> (L) // (P) <–> v perp n
   –> v (dot) n = 0
2. Empty set –> (L) // (P) but outside of (P)
   –> (L) // (P) <–> v perp n
   –> v (dot) n =0
3. Point –> (L) NOT // (P) and intersects (P)
   –> v (dot) n NOT EQUAL to 0

Question 15

Intersection of 2 (P)

Answer 15

1. Line –> (P1) NOT // (P2)
   –> n1 NOT // n2
2. Plane –> (P1) // (P2) + coincides (identical!)
   –> n1 // n2
3. Empty set –> (P1) // (P2) BUT DON’T coincide
   –> n1 // n2

Question 16

Direction vector of the line of intersection of 2 (P)

Answer 16

v = n1 x n2 (cross product)

Question 17

Intersection of 2 (L)

Answer 17

1. Empty set –> (L1) // (L2) BUT DON’T coincide
   v1 // v2
   –> (L1) NOT // (L2) + DON’T coincide
   (skew lines)
2. Line –> (L1) // (L2) and coincide (identical!)
   –> v1 // v2
3. Point –> (L1) NOT // (L2) BUT both in same (P)
   –> lie on same plane

Question 18

Distance between point to (P)

Answer 18

D = |axo + byo + czo + d|/sqrt[(a^2) + (b^2) + (c^2)]

Question 19

Steps distance between 2 // (P)

Answer 19

D ((P1), (P2)) = D ((A), (P1))

1. Find A = (xo, yo, zo) from (P2)
   –> set y, z = 0 and solve for x
2. Find (a, b, c) from (P1)
   –> n = (a, b, c)
3. D =|axo + byo + czo + d|/sqrt[(a^2) + (b^2) + (c^2)]

Question 20

Steps distance between (P) and // (L)

Answer 20

D ((L), (P)) = D ((A), (P))

1. Find A = (xo,yo,zo) from (L)
2. Find (a, b, c) from (P)
   –> n = (a, b, c)
3. D =|axo + byo + czo + d|/sqrt[(a^2) + (b^2) + (c^2)]

Question 21

Steps distance between point and (L)

Answer 21

1. Find AB from distance between point given and point on (L)
2. Find v from eq of line
   D = ||AB x v||/||v||

Question 22

Point on (P) closest to given point

Answer 22

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

Question 23

Point on (L) closest to given point

Answer 23

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

Question 24

What are skew lines?

Answer 24

(L1) NOT // (L2) and DO NOT intersect
–> on different planes

Question 25

Distance between skew lines

Answer 25

D = ||proj (v1 x v2) PQ||
= |PQ (dot) (v1 x v2)|/||v1 x v2||
–> tripe scalar product

Question 26

||U x V ||= ?

Answer 26

||U x V|| = ||U|| ||V|| sin(theta)

Question 27

Area of parallelogram

Answer 27

A = ||U x V||

Question 28

Triple scalar product

Answer 28

U (dot) (V x W) = V (dot) (W x U) = W (dot) (U x V)

Question 29

Volume of parallelepiped

Answer 29

V = U (dot) (V x W)

NOTE: if U (dot) (V x W) = 0
–> vectors don’t lie on the same plane