Objectives:
Graph vectors in 3 dimensions
Use cross products and dot products to find
perpendicular vectors
Vectors in 3D Space
Ordered triples
(x, y, z)
z
y
x
Locate the point (3, 5, 4)
Vectors
 Draw the vector from the origin to the point (3, -2, 6).
 Draw the vector from the origin to the point (-2, 5, 4)
 Find the ordered triple that represents the vector from
A(3, 7, -1) to B(10, -4, 0).
 (x2 – x1, y2 – y1, z2 – z1)
Magnitude
The force of the wind blowing the John Hancock
Building at one moment can be expressed as the vector
(3120 N, 195 N, 5 N). What is the magnitude of the
force?
|x|= ( x  x) 2  ( y  y ) 2  ( z  z ) 2
Add, Subtract, Multiply by a Scalar
Find an ordered triple to represent u in each equation if
v = (1, -3, -8) and w = (3, 9, -1).
u=v+w
u = 4v + 3w
u = 5v – 2w
Unit Vectors
 Rewrite a vector as the sum of unit vectors.
(3, -5, 8) =
i = (1, 0, 0)
j = (0, 1, 0)
k = (0, 0, 1)
 First, express GH as an ordered triple. Then write as
the sum of unit vectors for G(10, -3, 15) and H(4, 1, -11).
Practice
1.
2.
3.
4.
5.
Locate (3, 4, 9). Then find the magnitude of the
vector from the origin.
Locate (-2, 1, 3). Then find the magnitude of the
vector from the origin.
For A(-2, 5, 8) and B(3, 9, -3), find an ordered triple
that represents AB. Then find the magnitude of the
vector.
Write (9, 3, -2) as the sum of unit vectors.
If v = (1, -3, -8) and w = (3, 0, 1), find 4v – 3w.
Perpendicular Vectors
 Two vectors are perpendicular if and only if their inner
product is zero.
 2D a •b = x1x2 + y1y2
 3D a •b = x1x2 + y1y2 + z1z2
 Which of the following vectors are perpendicular?
a = (3, 12) b = (8, -2) c = (3, -2)
v = (-6, 2, 10) w = (4, 1, 3)
To find a perpendicular vector. . .
Cross Product
axb=
y1
z1
y2
z2
i
x1
z1
x2
z2
j
x1
y1
x2
y2
k
Given a = (5, 2, 3) and b = (-2, 5, 0), find the
cross product. Then verify that the vector is
perpendicular.
i
j k
axb= 5 2 3
2 5 0
Practice
1.
Find the inner product and state whether the vectors
are perpendicular.
(-6, 1) • (-1, 2)
(3, -2, 4) • (1, -4, 0)
2. Find the cross product. Then verify that the
resulting vector is perpendicular to the given vectors.
(1, 3, 2) x (2, -1, -1)
Assignment: 3D Worksheet