Lecture Note - Vectors in Three Dimensions
Michael Wang
1. Intro to the Three Dimensional Space
By convention, we label the positive side of the three dimension space by right-hand rule. Each
point in space has unique coordinates (a, b, c) relative to the axes. We usually use R3 to denote the
tree dimensional space. R3 is divided by the coordinate planes into eight octants.
Theorem 1. (Distance Formula in R3 ). The distance |P − Q| between the points P = (a1 , b1 , c1 )
and Q = (a2 , b2 , c2 ) is
p
|P − Q| = (a2 − a1 )2 + (b2 − b1 )2 + (c2 − c1 )2
The length or magnitude of v, denoted kvk, is the distance from P to Q.
P (x, y, z) and Q(a, b, c) are two points in R3 with length of R. If we regard Q as a fixed point in R3 ,
then a sphere of radius R can be defined. Thus by the distance formula, the coordinates of P (x, y, z)
must satisfy
p
(x − a)2 + (y − b)2 + (z − c)2 = R
Therefore, we can define the equations of spheres and cylinders
Definition 1. (Equations of Spheres and Cylinders). An equation of the sphere in R3 of radius
R centered at Q(a, b, c) is
(x − a)2 + (y − b)2 + (z − c)2 = R2
(1)
An equation of the right circular cylinder in R3 of radius R whose central axis is the vertical line
through (a, b, 0) is
(x − a)2 + (y − b)2 = R2
(2)
2. Vectors in R3
Now, let us extend our two dimensional space vector knowledge to the three dimensional space.
Definition 2. A three-dimensional vector, v is determined by two points in R3 : an initial point
P = (a1 , b1 , c1 )(tail) and a terminal point Q = (a2 , b2 , c2 )(head). We write
−−→
v = PQ
The length or magnitude of v, denoted kvk, is the distance from P to Q.
p
−−→
kvk = kP Qk = (a2 − a1 )2 + (b2 − b1 )2 + (c2 − c1 )2
The terminology and basic properties discussed in R2 carry over to R3 with little change.
• Translation
• Equivalent
• Parallel
1
• Position vector
• Components - Two vectors are equivalent iff they have the same components.
• Vector addition and scalar multiplication
In terms of components, if v = ha1 , b1 , c1 i and w = ha2 , b2 , c2 i, then
λv = λha1 , b1 , c1 i = hλa1 , λb1 , λc1 i
v + w = ha1 , b1 , c1 i + ha2 , b2 , c2 i = ha1 + a2 , b1 + b2 , c1 + c2 i
• Standard basis vectors in R3 are
i = h1, 0, 0i
j = h0, 1, 0i
k = h0, 0, 1i
• Linear combination - Every vector is a linear combination of the standard basis vectors
ha, b, ci = ah1, 0, 0i + bh0, 1, 0i + ch0, 0, 1i = ai + bj + ck
• Normalization
ev =
v
kvk
3. Parametric Equations of a Line
A line in R2 is defined by a single linear equation such as y = mx + b. In R3 , a single linear equation
defines a plane rather than a line. Therefore, we describe lines in R3 in parametric form.
Definition 3. (Equation of a Line (Point-Direction Form)). The line L through P0 = (x0 , y0 , z0 )
in the direction of v = ha, b, ci is described by
Vector parametrization:
−−→
r(t) = OP0 + tv = hx0 , y0 , z0 i + tha, b, ci
(3)
Parametric equations:
x = x0 + at,
y = y0 + bt,
z = z0 + ct
(4)
The vector v is called a direction vector for L.
Different Parametrizations of the Same LIne, Intersection of Two Lines.
Definition 4. (Line through Two Points). The line through P = (a1 , b1 , c1 ) and Q = (a2 , b2 , c2 )
is described by
Vector parametrization:
−−→
−−→
r(t) = (1 − t)OP + tOQ = (1 − t)ha1 , b1 , c1 i + tha2 , b2 , c2 i
(5)
Parametric equations:
x = a1 + (a2 − a1 )t,
y = b1 + (b2 − b1 )t,
z = c1 + (c2 − c1 )t
(6)
for −∞ < t < ∞. This parametrization traces the segment P Q from P to Q at t varies form 0 to 1.
2 b1 +b2 c1 +c2
The midpoint of P Q corresponds to t = 21 : r( 12 ) = ( a1 +a
2 ,
2 ,
2 ).
2