Lines and Planes (review)

Question 1

Is the Vector Parametric construction of a line unique?

Answer 1

no

Question 2

Describe a Vector Parametric construction of a line L.

Answer 2

L = {v0+td | t∈R} where

v0: Position vector (any point on L)
d: Direction vector (non-zero)

Question 3

Describe parametric form.

Answer 3

(x,y,z) = (a0,b0,c0) + t(a1,b1,c1) + s(a2,b2,c2),

a0,b0,c0) is the position vector and (a1,b1,c1) and (a2,b2,c2
are the direction vectors. t,s∈R are the parameters

Question 4

Describe Cartesian form.

Answer 4

In R^3, a plane is defined as the set of all points (x,y,z), such that
ax+by+cz = d,
a,b,c,d∈R,

n = (a,b,c) is a normal vector to the plane.

Question 5

Describe Normal form.

Answer 5

A plane P in R^3 through v0 with normal vector n is given by

P := {v∈R^3 | (v−v0) . n = 0},
where v represents a point on P and v0 is the position vector.

Question 6

What is a normal vector?

Answer 6

A non-zero vector and is orthogonal to all

vectors on the plane.

Question 7

Describe the cross product (determinant)

Answer 7

Given a,b,c,d∈R, let’s define (matrix)
|a b|
|c d|
:= ad − bc (determinant)

Question 8

Define the cross producct in R^3.

Answer 8

Given x=(x1,x2,x3) and y=(y1,y2,y3) in R^3, their cross
product x × y is given by
x × y := 
( |x2 x3|, _|x1 x3|, |x1 x2| )
( |y2 y3|    |y1 y3|  |y1 y2| )

Question 9

Define the cross producct in R^3 based on unit vectors.

Answer 9

Let ˆi=(1, 0, 0), ˆj=(0, 1, 0), k=(0, 0, 1). Then,
x × y :=
|ˆi   ˆj   kˆ|
|x1 x2 x3| = 
|y1 y2 y3|

|x2 x3|ˆi-|x1 x3|^j+|x1 x2|^k
|y2 y3| |y1 y3| |y1 y2|

Question 10

What are the properties of cross product?

Answer 10

1. x×y = −y×x, for x,y∈R^3,
2. (x×y).x = (x×y).y = 0,
3. (x+y)×z = x×z + y×z, for x,y,z∈R^3,
4. x×(y×z) ≠ (x×y)×z (in general).
   ||x×y||= ||x|| ||y|| sin(α), where 0≤α≤π is the angle
   between x and y. It is the area of parallelogram with x and y

Question 11

What is the first application of cross products?

Answer 11

The cross product gives a normal vector to the plane that is

parallel to the direction vectors x and y

Question 12

What is the second application of cross products?

Answer 12

Given u,v,w∈R^3, the volume of a parallelepiped in R^3 with
u,v,w as sides is given by the scalar triple product
|(u×v). w|

Question 13

What are the three remarks with regards to Cross products in R^3?

Answer 13

1. (u×v).w = (w×u).v = (v×w).u = etc. (with the same
   cyclic order),
2. Three vectors u,v,w are co-planar ⇔ (u×v).w = 0,
3. If u=(u1,u2,u3), v=(v1,v2,v3) and w=(w1,w2,w3), then
   u.(v×w) =
   |u1 u2 u3|
   |v1 v2 v3|
   |w1 w2 w3|