Vectors

Question 1

Vector

Definition

Answer 1

a quantity with both magnitude and direction

Question 2

Scalar Multiplication of Vectors

Answer 2

v = a vector
r = a scalar
-for r>0, rv has the same direction as v
-for r

Question 3

What is a unit vector?

Answer 3

vectors for which ||v|| = 1

Question 4

Addition of Vectors

Answer 4

-the parallelogram rule
OP = (p1 ,p2, p3)
OQ = (q1, q2, q3)
OP + OQ 
= (p1+q1, p2+q2, p3+q3)

Question 5

The Scalar Product

Definnition

Answer 5

a.b = ||a|| * ||b|| * cosθ

where θ is the angle between the two vectors a and b

Question 6

Is the scalar product commutative?

Answer 6

yes

a.b = b.a

Question 7

Scalar Product

a and b perpendicular

Answer 7

cosθ = 0

so a.b = 0

Question 8

Scalar Product

a and b in the same direction

Answer 8

cosθ = 1
a.b = ||a||*||b||

Question 9

Scalar Product

a and b in the opposite direction

Answer 9

cosθ = -1
a.b = -||a||*||b||

Question 10

Scalar Products of the Unit Vectors

Answer 10

i. i = j.j = k.k = 1

i. j = i.k = j.k = 0

Question 11

Norm

Definition and Notations

Answer 11

the norm of a:

||a|| = √(a1² +a2² +a3²)

Question 12

The Scalar Product

Coordinate Form

Answer 12

a.b = a1b1 + a2b2 + a3b3
where a = (a1i, a2j, a3k)
and b = (b1i, b2j, b3k)

Question 13

The Vector Product

Answer 13

||axb|| = ||a||||b||sinθ

where θ is the angle between a and b

Question 14

Comparison of the Scalar and Vector Products

Answer 14

• in contrast to the scalar product, the vector product of two vectors is another vector
• the direction of this vector is not completely determined since there are two options for the vector to be perpendicular to the two given vectors
• in order to determine the direction of the vector product, use the right hand rule

Question 15

The Right Hand Rule

Answer 15

• with a flat hand point your fingers in the direction of the first vector
• bend your fingers at the knuckles so that they point in the same direction as the second vector
• the direction that your thumb now points in is the direction of the vector product

Question 16

Vector Product

a and b are parallel

Answer 16

θ = 0
so ||axb|| = 0
the only vector with 0 length is the zero vector so we can concude that axb = 0

Question 17

Is the vector product commutative?

Answer 17

no

axb = -bxa

Question 18

The Vector Product

Coordinate Form

Answer 18

(a1, a2, a3) x (b1, b2, b3) = (a2b3-a3b2, a3b1-a1b3, a1b2-a2b1)

Question 19

What sets of information can be used to describe a line?

Answer 19

1) a point P and a vector v, determine a unique line through P in the direction of v
2) two points P1 and P2, determine the line connecting them
3) in ℝ² a line is determined by its slope and a point P through which it goes

Question 20

Equations for Lines

-Point P and a vector v

Answer 20

-let Q be an arbitrary point on the line
OQ = OP + sv, sℝ
-OQ denotes the position vector of Q
-in ℝ^3 
(x,y,z) = (P1,P2,P3) + s*(v1,v2,v3)
x = p1 + s*v1
y = p2 + s*v2
z = p3 + s*v3
-can rearrange for s and set these equations equal to each other
s=(x-P1)/v1 = (y-P2)/v2 = (z-P3)/v3

Question 21

Equations for Lines

-two points P1 and P2

Answer 21

-if we know two points P1 and P2 are on the line then:
P1P2 = OP2 - OP1
is parallel to the line
OQ = OP1 + s*P1P2
-where Q is an arbitrary point on the line

Question 22

Equations for Lines

-in ℝ², slope of the line and a point P

Answer 22

-if P=(p1,p2) and Q=(q1,q2) then the slope of the line is 
m = (q2-p2)/(q1-p1)
(x,y) = (p1,p2) 
\+ s*(1,m)
OR
x = p1 + s
y = p2 + s*m

Question 23

What sets of information can be used to characterise planes in ℝ³?

Answer 23

1) a point P and two vectors v1 and v2, determine a plane containing P and parallel to v1 and v2
2) three points P1, P2 and P3, determine a plane that contains all 3 points as long as the points do not lie in a straight line
3) a point P and a vector n, determine the plane which goes through P through which all vectors in the plane are perpendicular to n

Question 24

Equations for Planes

-one point P and two vectors v1 and v2

Answer 24

OQ = OP + s*v1 + t*v2
s,t = ℝ

Question 25

Equations for Planes

-3 point, P1, P2, P3

Answer 25

OQ=OP1+sP1P2+tP1P3

Question 26

Equations of Planes

-point p and a vector n perpendicular to the plane

Answer 26

-if n is perpendicular to the plane then all vectors in the plane are perpendicular to n
-let Q be a point on the plane Q=(x,y,z)
PQ.n = 0
PQ = OQ - OP
so, (OQ - OP).n=0
OQ.n = OP.n
n1x + n2y + n3z = n1p1 + n2p2 + n3p3

Question 27

Normal Vector to the Plane

Definition

Answer 27

if a vector n is perpendicular to a plane S, we say that n is a normal vector to the plane S

Question 28

Normal Vector for a Plane in the Form

a1x + a2y + a3z = b

Answer 28

n = (a1,a2,a3)
-consider a point P in the plane
a1p1+a2p2+a3p3=b
-n is a normal vector if PQ.n=0
(OQ-OP).n = 0
OQ.n = OP.n
(x,y,z).n = (p1,p2,p3).n

Question 29

Equation for a Circle in ℝ²

Answer 29

(x-x1)² + (y-y1)² = r²
(x1,y1) = centre
r = radius
-this can be rewritten as
||(x,y) - (x1,y1)||² = r²

Question 30

Equation for a sphere in ℝ³

Answer 30

||(x,y,z) - (x1,y1,z1)||² = r²
which means:
(x-x1)²+(y-y1)²+(z-z1)²=r²