Vectors Flashcards
Vector
Definition
a quantity with both magnitude and direction
Scalar Multiplication of Vectors
v = a vector
r = a scalar
-for r>0, rv has the same direction as v
-for r
What is a unit vector?
vectors for which ||v|| = 1
Addition of Vectors
-the parallelogram rule OP = (p1 ,p2, p3) OQ = (q1, q2, q3) OP + OQ = (p1+q1, p2+q2, p3+q3)
The Scalar Product
Definnition
a.b = ||a|| * ||b|| * cosθ
where θ is the angle between the two vectors a and b
Is the scalar product commutative?
yes
a.b = b.a
Scalar Product
a and b perpendicular
cosθ = 0
so a.b = 0
Scalar Product
a and b in the same direction
cosθ = 1 a.b = ||a||*||b||
Scalar Product
a and b in the opposite direction
cosθ = -1 a.b = -||a||*||b||
Scalar Products of the Unit Vectors
i. i = j.j = k.k = 1
i. j = i.k = j.k = 0
Norm
Definition and Notations
the norm of a:
||a|| = √(a1² +a2² +a3²)
The Scalar Product
Coordinate Form
a.b = a1b1 + a2b2 + a3b3
where a = (a1i, a2j, a3k)
and b = (b1i, b2j, b3k)
The Vector Product
||axb|| = ||a||||b||sinθ
where θ is the angle between a and b
Comparison of the Scalar and Vector Products
- 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
The Right Hand Rule
- 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
Vector Product
a and b are parallel
θ = 0
so ||axb|| = 0
the only vector with 0 length is the zero vector so we can concude that axb = 0
Is the vector product commutative?
no
axb = -bxa
The Vector Product
Coordinate Form
(a1, a2, a3) x (b1, b2, b3) = (a2b3-a3b2, a3b1-a1b3, a1b2-a2b1)
What sets of information can be used to describe a line?
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
Equations for Lines
-Point P and a vector v
-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
Equations for Lines
-two points P1 and P2
-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
Equations for Lines
-in ℝ², slope of the line and a point P
-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
What sets of information can be used to characterise planes in ℝ³?
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
Equations for Planes
-one point P and two vectors v1 and v2
OQ = OP + s*v1 + t*v2 s,t = ℝ
Equations for Planes
-3 point, P1, P2, P3
OQ=OP1+sP1P2+tP1P3
Equations of Planes
-point p and a vector n perpendicular to the plane
-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
Normal Vector to the Plane
Definition
if a vector n is perpendicular to a plane S, we say that n is a normal vector to the plane S
Normal Vector for a Plane in the Form
a1x + a2y + a3z = b
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
Equation for a Circle in ℝ²
(x-x1)² + (y-y1)² = r² (x1,y1) = centre r = radius -this can be rewritten as ||(x,y) - (x1,y1)||² = r²
Equation for a sphere in ℝ³
||(x,y,z) - (x1,y1,z1)||² = r²
which means:
(x-x1)²+(y-y1)²+(z-z1)²=r²
