Math 152 Class Notes
December 3, 2015
11.2 Vectors and the Dot Product in Three Dimensions
In this section, we study vectors in three dimensions. The term vector is used to
indicate a quantity that has both magnitude and direction.
Denition. A three-dimensional vector is an ordered triple a = ha1, a2, a3i of real
numbers. The number a1 , a2 and a3 are called the components of a.
We denote a vector by printing a letter in boldface a or by putting an arrow above the
letter ~a. We use the notation ha1 , a2 , a3 i for the ordered triple that refers to a vector
so as not to confuse it with (a1 , a2 , a3 ) the ordered triple that refers to a point in R3 .
Geometrically, a vector is represented by an arrow or a directed line segment. The
starting point of a vector does not have to be the origin. If the vector a = ha1 , a2 , a3 i
starts at the point A(x, y, z), then this vector can be identied as the directed line
−→
segment from A(x, y, z) to the point B(x + a1 , y + a2 , z + a3 ). This segment AB
is called a representation of the vector a. A particular representation of a is the
−→
directed segment OP from the origin O to the point P (a1 , a2 , a3 ).
−→
Given A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ), the vector a with representation AB is
a = hx2 − x1 , y2 − y1 , z2 − z1 i
Example 1. Find the vector represented by the directed line segment with initial point
A(2, −3, 4) and terminal point B(−2, 1, 1).
The magnitude or length of the vector a is the length of any of its representations
and is denoted by |a|.
The length of the three-dimensional vector a = ha1 , a2 , a3 i is
q
|a| = a21 + a22 + a23
The zero vector, denoted by 0, has length 0. It is the only vector with no specic
direction.
A unit vector is a vector whose length is 1. For instance, h1, 0, 0i is a unit vector.
In general, if a 6= 0 then the unit vector that has the same direction as a is
u=
a
1
a=
|a|
|a|
Example 2. (a) Find the magnitude of the vector with initial point A(2, −3, 4) and
terminal point B(−2, 1, 1).
−→
(b) Find a unit vector in the direction of the vector AB .
Vector Addition and Scalar Multiplication
If a = ha1 , a2 , a3 i, b = hb1 , b2 , b3 i and c is a scalar, then
a + b = ha1 , a2 , a3 i + hb1 , b2 , b3 i = ha1 + b1 , a2 + b2 , a3 + b3 i
a − b = ha1 , a2 , a3 i − hb1 , b2 , b3 i = ha1 − b1 , a2 − b2 , a3 − b3 i
ca = cha1 , a2 , a3 i = hca1 , ca2 , ca3 i
The following three vectors i, j, and k are called the standard basis vectors.
i = h1, 0, 0i,
j = h0, 1, 0i,
k = h0, 0, 1i
They are unit vectors in the directions of the positive x-, y -, and z -axes. Any vector
in R3 can be expressed in terms of i, j, and k.
a = ha1 , a2 , a3 i = a1 i + a2 j + a3 k
Example 3. Given a = h4, 0, 3i and b = i + 2j − 3k, nd a vector of length 3 in the
direction of the vector 2a − b.
The Dot Product
Denition. The dot product of two vectors a and b is the number
a · b = |a||b| cos θ
where θ is the angle between the vectors a and b, 0 ≤ θ ≤ π .
Based on the Law of Cosines, we can compute the dot product of two vectors a =
ha1 , a2 , a3 i and b = hb1 , b2 , b3 i in terms of their components.
a · b = a1 b 1 + a2 b 2 + a3 b 3
Example 4. Find the dot product of a = h2, 2, −1i and b = h5, −3, 2i
As a direct consequence of the denition of the dot product, we have the following
formula to compute the angle θ between two nonzero vectors a and b.
cos θ =
a·b
|a||b|
Example 5. Given the points A(5, 5, 1), B(3, 3, 2) and C(1, 4, 4), nd ∠ABC .
Two nonzero vectors are called perpendicular or orthogonal if the angle between
π
them is . The zero vector 0 is considered to be perpendicular to all vectors. The dot
2
product can be used to determined whether two vectors are orthogonal.
Two vectors a and b are orthogonal if and only if a · b = 0
Example 6. Show that a = 2i + 2j − k is perpendicular to b = 5i − 4j + 2k.
Projection
Given two vectors a and b, we can nd the vector projection of b onto a denoted
by proja b:
The scalar projection of b onto a (also called the component of b along a) is
dened to be the signed magnitude of the vector projection,
The projection can be computed by taking the dot product of a and b.
Scalar projection of b onto a:
Vector projection of b onto a:
a·b
|a|
a
a·b a
a·b
proja b = (compa b)
=
=
a
|a|
|a| |a|
|a|2
compa b = |b| cos θ =
Example 7. Find the scalar projection and vector projection of b = h1, 1, 2i onto
a = h−2, 3, 1i