Chapter 3: Vector Spaces

Question 1

Theorem 3.2.1

Answer 1

If v is a vector in Rn, and if k is any scalar, then: - ||v|| ≥ 0 - ||v|| = 0 iff v = 0 - ||kv|| = |k|.||v||

Question 2

Unit vector

Answer 2

A vector of norm 1.

u = ¹/_{||<strong>v</strong>||} v

Question 3

Standard unit vectors

Answer 3

The unit vectors in the positive directions of the coordinate axes when a rectangular coordinate system is introduced in R² or R³.

Question 4

Dot product of u and v

Answer 4

If u and v are nonzero vectors in R2 or R3, and if θ is the angle between u and v, then the dot product denoted by u•v is defined as:

u•v = ||u|| ||v|| cosθ

Question 5

Euclidean inner product (dot product)

Answer 5

If u = (u₁, u₂, … u_n) and v = (v₁, v₂, … v_n) are vectors in Rⁿ, then the dot product is denoted by u • v and is defined by: u • v = u₁v₁ + u₂v₂ + … + u_nv_n

Question 6

Express the length of a vector in terms of a dot product

Answer 6

||v|| = √(v•v)

Question 7

Triangle inequality for vectors

Answer 7

||u** + **v **|| ≤ ||u *|| + ||v ***||

Question 8

Triangle inequality for distances

Answer 8

d(u, v) ≤ d(u, w) + d(w, v)

Question 9

Parallelogram Equation for vectors

Answer 9

||u + v||² + ||u - v||² = 2( ||u||² + ||v||²)

Question 10

The angle θ between 2 nonzero vectors u and v in Rⁿ

Answer 10

θ = cos^-1( ^{<strong>u•v</strong>}/_{(||<strong>u</strong>|| ||<strong>v</strong>||)} )

Question 11

Orthogonal

Answer 11

Two nonzero vectors u and v in Rⁿ are said to be orthogonal if u•v = 0

Question 12

Equivalent vectors

Answer 12

Vectors with the same length and direction.

Question 13

Length of v

||v||

Answer 13

√v₁² + v₂² + … + v_n²

Question 14

Cauchy-Schwarz Inequality

Answer 14

|u•v| ≤ ||u|| ||v||

Question 15

Orthogonal projection of u on a

A.k.a. the vector component of u ALONG a

Answer 15

proj_au = ( ^{<strong>u•a</strong>}/||a||² ) a

Question 16

Vector component of u ORTHOGONAL TO a

Answer 16

u - proj_au

Question 17

|| proj**_au **||

Answer 17

^{|<strong>u•a</strong>|}/_{||<strong>a</strong>||}

Question 18

Theorem of Pythagoras in Rⁿ

Answer 18

If u and v are orthogonal vectors in Rⁿ with the Euclidean inner product, then ||u + v||² = ||u||² + ||v||²

Question 19

Distance D between the point P0(x0, y0) and the line ax + by + c = 0 in R2

Answer 19

D = |ax0 + by0 + c|/sqrt(a^2 + b^2)

Question 20

Distance D between the point P0(x0, y0) and the plane ax + by + cz + d = 0 in R3

Answer 20

D = |ax0 + by0 + cz0 + d|/sqrt(a^2 + b^2 + c^2)