Chapter 3: Euclidean vector spaces

Question 1

||kv||

Answer 1

= |k| ||v||

Question 2

Angle between 2 vectors using cos

Answer 2

cosø = ^{<strong>u•v</strong>}/_{||<strong>u</strong>|| ||<strong>v</strong>||}

Question 3

Cauchy-Schwarz Inequality

Answer 3

|u•v| ≤ ||u|| ||v||

Question 4

Parallelogram Equation for Vectors

Answer 4

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

Question 5

Euclidian inner product:

Answer 5

u•v = ¹/₄||u + **v||² - ¹/₄ ||u** - v||²

Question 6

Link between dot-product multiplication by matrix A and A transpose:

Answer 6

*A *u • **v **= **u • **A^T v

u • Av = A^T u • v

Question 7

A normal to a line

ax + by + c = 0

Answer 7

**n **= (a, b)

Question 8

Homogeneous equations in 2 or 3 unknowns can be written in vector form:

Answer 8

**n • x ** = 0

where n is the vector of coefficients

x is the vector of unknowns

Question 9

proj_{<strong>a</strong>}**v **

Answer 9

( v•u )u

where u = ^{<strong>a</strong>}/_||a||, the unit vector of a.

Question 10

If **u **and v are orthogonal vectors in Rⁿ with the Euclidean inner product, then

Answer 10

||u + v||² = ||u||² + ||v||²

Question 11

Distance between the Point P(x₀, y₀) and the line ax + by + c = 0

Answer 11

D =

Question 12

||proj_{<strong>a</strong>}u||

Answer 12

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

_{= ||<strong>u</strong>|| |cosø|}

Question 13

Line through x₀, parallel to v

Answer 13

x = x₀ + *t *v

Question 14

Plane through x₀, parallel to v₁ and v₂.

Answer 14

x = x₀ + t₁ v₁ + t₂ v₂

Question 15

Line segment from **x₀ **to x₁.

Answer 15

x = **x₀ + t** **(x₁** - x₀)

( 0 ≤ t ≤ 1 )

Question 16

All vectors in Rⁿ that are orthogonal to every row vector of a matrix A.

Answer 16

Solution set to:

Ax = 0

Question 17

The general solution of a consistent linear system Ax = b, using the general solution of Ax = 0.

Answer 17

Obtained by adding any specific solution of Ax = b to the general solution of Ax = 0.

Question 18

4 Relationships involving Cross Product and Dot product:

Answer 18

• u • (** u** x v ) = 0 (u x v is orthogonal to u. (and to v))
• || u x v ||² = ||u||²||v||² - (u•v)²
• u x (** v** x w ) = ( u • **w )v** - ( u • **v )w**
• ( u x v ) x **w = ( u • **w )v - ( v • w )u

Question 19

3 properties of the cross product

Answer 19

• u x v = - (v x u)
• k(u x v) = (ku) x v = u x (kv)
• u x u = 0

Question 20

Angle between two vectors in terms of sin

Answer 20

sinø = ^{||<strong>u</strong>x <strong>v</strong>||}/_{||<strong>u</strong>|| ||<strong>v</strong>||}

Question 21

|| u x v || in 3 space geometry, equals

Answer 21

the area of the parallelogram determined by u and v.

Question 22

Scalar triple product of u, v, and w:

Answer 22

u • ( v x w )

( = w • ( u x v ) )

( = v • ( w x u ) )

Question 23

The absolute value of the determinant:

Answer 23

Equals the area of the parallelogram in 2-space determined by the vector u = (u₁, u₂) and v = (v₁, v₂)

Question 24

The absolute value of the determinant:

(in geometric terms)

Answer 24

Equals the volume the parallelepiped in 3-space determined by the vector u, v and w.

Question 25

If the vectors u, v and w have the same initial point, then they ly in the same plane iff

Answer 25

The scalar triple product = 0

u • (v x w) = 0