LECTURE 2: PLANES AND SURFACES
MINGFENG ZHAO
January 07, 2015
Points and vectors in 3 dimensions
Definition 1. The set of all points described by the triples (x, y, z) is called the three-dimensional space, or xyz-space,
or R3 . The element in R3 is called a point.
Figure 1. A point in R3
Definition 2. The distance from the point P (x1 , y1 , z1 ) to the point Q(x2 , y2 , z2 ) is the length of the line segment P Q
and given by:
p
(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
Definition 3. The vector from the point P (x1 , y1 , z1 ) to the point Q(x2 , y2 , z2 ) is defined by:
−−→
P Q = hx2 − x1 , y2 − y1 , z2 − z1 i.
1
2
MINGFENG ZHAO
Definition 4. The length or magnitude of a vector u = hu1 , u2 , u3 i is defined by:
|u| =
q
u21 + u22 + u23 .
Remark 1. The distance from the point P (x1 , y1 , z1 ) to the point Q(x2 , y2 , z2 ) is the length or magnitude of the vector
−−→
P Q.
Remark 2. The following are some differences between points and vectors:
Points
Vectors
Points are elements in R3
Vectors are NOT elements in R3
Points use ‘( , , )’
Vectors use ‘h , , i’
Points mean the location in R3
Vectors focus on the direction
The distance from one point to another point
The length or magnitude of a vector
Dot products of vectors
Given vectors u = hu1 , u2 , u3 i, v = hv1 , v2 , v3 i and w, a real number c, then
• There holds that
u·v :
=
|u||v| cos(θ)
=
u1 v1 + u2 v2 + u3 v3
u·v
−1
θ = cos
|u||v|
√
|u| =
u·u
• Properties of dot products:
– Commutative property: u · v = v · u.
– Associative property: c(u · v) = (cu) · v = u · (cv).
– Distributive property: u · (v + w) = u · v + u · w.
Example 1 (Parallelogram Law). For the parallelogram ABCD, show that |BD|2 + |AC|2 = 2[|AB|2 + |BC|2 ].
−−→
−−→
−−→
−→
−−→
−→
Let u = DC and v = DA, then DB = u + v and CA = u − v. Notice that |BD| = |DB|, |AC| = |CA|,
−−→
−−→
−−→
−−→
|AB| = |AB| = |DC| and |BC| = |BC| = |DA|. Therefore, we only need to show that |u + v|2 + |u − v|2 = 2[|u|2 + |v|2 ].
LECTURE 2: PLANES AND SURFACES
3
By the definition of the magnitude, we have
|u + v|2
=
(u + v) · (u + v)
=
(u + v) · u + (u + v) · v
By the distributive property
= u·u+v·u+u·v+v·v
= |u|2 + 2u · v + |v|2
|u − v|2
By the commutative property
=
(u − v) · (u − v)
=
(u − v) · u − (u − v) · v
By the distributive property and associative property
= u·u−v·u−u·v+v·v
= |u|2 − 2u · v + |v|2
By the distributive property
By the distributive property and associative property
By the commutative property.
Then we get
|u + v|2 + |u − v|2 = 2[|u|2 + |v|2 ].
Equations of planes
Definition 5. Given two vectors u and v, we say that u is parallel to v if there exists some nonzero constant λ such
that u = λv.
−−→
Definition 6. Given a fixed point P0 in R3 and a nonzero vector n, the set of all points P in R3 for which P0 P is
orthogonal to n is called a plane. In this case, this nonzero vector n is called the normal vector of this plane.
4
MINGFENG ZHAO
Theorem 1. The plane passing through the point P0 = (x0 , y0 , z0 ) with a nonzero normal vector n = ha, b, ci is described
by the equation:
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0,
or
ax + by + cz = d,
where d = ax0 + by0 + cz0 .
Remark 3. The general equation for a plane is given by ax + by + cz = d, then a normal vector can be n = ha, b, ci.
Any nonzero multiple of n is also a normal vector to this plane. In particular, there are infinitely many normal vectors
to one plane, and those normal vectors are parallel to each other.
Moreover, for any nonzero constant λ, the equations λax + λby + λcy = λd and ax + by + cz = d give the same plane,
that is, a plane may have different equations.
Example 2. Find an equation of the plane passing through P0 (2, −3, 4) with a normal vector n = h−1, 2, 3i.
By Theorem 1, we have
(−1) · (x − 2) + 2 · (y + 3) + 3 · (z − 4) = 0.
That is,
−x + 2 + 2y + 6 + 3z − 12 = 0.
So we get
−x + 2y + 3z = 4.
Parallel and orthogonal planes
Definition 7. For any two distinct planes P and Q with normal vectors n and m respectively, the angle between these
two planes P and Q is defined as the acute angle between n and m (i.e., between 0 and π/2).
Definition 8. ˙
I. Two distinct planes are parallel if their respective normal vectors are parallel (that is, the normal vectors are
scalar multiples of each other, or the angle is either 0 or π).
II. Two distinct planes are orthogonal if their respective normal vectors are orthogonal (that is, the dot product of
the normal vectors is zero, or the angle is
π
2 ).
LECTURE 2: PLANES AND SURFACES
5
Figure 2. Two parallel planes
Figure 3. Two orthogonal planes
Example 3. Find an equation of the plane Q that passes through the point (−2, 4, 1) is parallel to the plane R :
3x − 2y + z = 4.
Since the plane Q is parallel to the plane R : 3x − 2y + z = 4, then we can take the normal vector n of Q to be
n = h3, −2, 1i. Since Q passes through the point (−2, 4, 1), then the equation of Q is
3 · (x + 2) + (−2) · (y − 4) + 1 · (z − 1) = 0.
That is,
3x + 6 − 2y + 8 + z − 1 = 0.
6
MINGFENG ZHAO
So we get
3x − 2y + z = −13.
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca