Vector Geometry (review)

Question 1

What is a set?

Answer 1

A collection of objects, where it is often useful to use the
following notation:
{Objects | logical predicates}

Question 2

Define the aspects of {Objects | logical predicates}.

Answer 2

{ } = The set of all
Objects = variables
| = Such that
Log. Pre. = rules

Question 3

What does the set notation of a plane look like?

Answer 3

R^2 = {(x, y)| x, y ∈ R}

Question 4

What does the set notation of a vector in 3-space look like?

Answer 4

R^3 = {(x, y, z)| x, y, z ∈ R}

Question 5

What does the set notation of a vector in n-space look like?

Answer 5

R^n = = {(x1, x2, · · · , xn)| x1, x2, · · · , xn ∈ R}

Question 6

Give the definition of linear combinations.

Answer 6

If k1,···,kn∈R are scalars and u1,···,un∈R^n
are
vectors, then
k1u1 +···+ knun
is called a linear combination of u1, · · · , un.

Question 7

What are the 10 Algebraic properties of vector addition, scalar multiplication:
for x,y,z∈R^n
and a,b∈R?

Answer 7

• Closure under Addition
• Closure under Scalar Multiplication
• Additive Identity
• Additive Inverse
• Commutativity of Addition
• Associativity of Addition
• Distributivity of scalars onto vectors
• Distributivity of Vectors onto scalars
• Associativity of Multiplication
• Multiplicative Identity

Question 8

Define the Inner dot product

Answer 8

x . y = x1y1 +···+ xnyn ∈ R

Question 9

Define the norm (length)

Answer 9

||x|| := √(x.x)

sqrt of dot prod

Question 10

What are the 8 properties of Dot Product?

Answer 10

If x,y,z∈R^n
and a∈R. Then
1. ||x|| ≥ 0
2.  ||x||= 0 ⇔ x = 0
3.  ||ax||= |a| ||x||
4. ||x-y|| = ||y-x||
5. x.y = y.x
6.  (ax).y = a(x.y) = x.(ay)
7.  (x+y).z = x.z + y.z
8. x.(y+z) = x.y + x.z

Question 11

When is x.y = 0?

Answer 11

When x and y are orthogonal

Question 12

Define orthogonality.

Answer 12

If x,y∈R^n, then x and y are said to be orthogonal if

x . y = 0.

Question 13

Define the Cauchy-Schwarz Inequality.

Answer 13

For x,y∈R^n, we have

|x.y| ≤ ||x|| ||y||, (x.y∈R)

Question 14

Define Triangle Inequality.

Answer 14

||x+y|| ≤ ||x||+||y||

Question 15

Define the angle between vectors in R^n.

Answer 15

If x,y∈R^n and x,y≠0, then the angle θ between x and y is defined by
cos(θ) := x.y/(||x|| ||y||),
such that 0 ≤ θ ≤ π

Question 16

When are two vectors parallel?

Answer 16

1. If the angle between them is either 0 or π.

Question 17

Define orthogonal projections.

Answer 17

Suppose x and y are non-zero vectors in R^n. The projection of y
onto x (denoted by Projx(y)) is the unique vector satisfying:
- Projx(y) is a scalar multiple of x (parallel to x),
- y−Projx(y) is orthogonal to x (their dot product vanishes)

Question 18

How do you find orthogonal projections?

Answer 18

Projx(y) := (x.y/(||x||^2))x

Question 19

When are two vectors u and v collinear (parallel)?

Answer 19

If there exists scalars a,b∈R (not both zero), such that au + bv = 0.

Question 20

When are three vectors u, v, and w coplanar?

Answer 20

If there exist scalars

a,b,c∈R (not all zero) such that au + bv + cw = 0.