Download Section 11.3 The Cross Product of Two Vectors

Section 11.3
•
The Cross Product of Two Vectors
Name______________________________________________
Section 11.3
The Cross Product of Two Vectors
Objective: In this lesson you learned how to find cross products of
vectors in space, use geometric properties of the cross
product, and use triple scalar products to find volumes of
parallelepipeds.
I. The Cross Product (Pages 827−828)
A vector in space that is orthogonal to two given vectors is called
their
cross product
.
Let u = u1i + u2j + u3k and v = v1i + v2j + v3k be two vectors in
space. The cross product of u and v is the vector
u×v=
(u2v3 − u3v2)i − (u1v3 − u3v1)j + (u1v2 − u2v1)k
Describe a convenient way to remember the formula for the
cross product.
Answers will vary.
Example 1: Given u = − 2i + 3j − 3k and v = i − 2j + k, find
the cross product u × v.
− 3i − j + k
Let u, v, and w be vectors in space and let c be a scalar.
Complete the following properties of the cross product:
1. u × v =
− (v × u)
2. u × (v + w) =
(u × v) + (u × w)
3. c(u × v) =
(cu) × v = u × (cv)
4. u × 0 =
0×u=0
5. u × u =
0
6. u • (v × w) =
(u × v) • w
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE
Copyright © Houghton Mifflin Company. All rights reserved.
What you should learn
How to find cross
products of vectors in
space
191
192
Chapter 11
•
Analytic Geometry in Three Dimensions
II. Geometric Properties of the Cross Product
(Pages 829−830)
What you should learn
How to use geometric
properties of cross
products of vectors in
space
Complete the following geometric properties of the cross
product, given u and v are nonzero vectors in space and θ is the
angle between u and v.
1. u × v is orthogonal to
both u and v
|| u || || v || sin θ
2. || u × v || =
3. u × v = 0 if and only if
.
.
u and v are scalar multiples
4. || u × v || = area of the parallelogram having
adjacent sides
u and v as
.
III. The Triple Scalar Product (Page 831)
For vectors u, v, and w in space, the dot product of u and v × w
is called the
.
triple scalar product
of u, v, and w, and
What you should learn
How to use triple scalar
products to find volumes
of parallelepipeds
is found as
u1
u2
u3
u • ( v × w ) = v1
v2
v3
w1 w2 w3
The volume V of a parallelepiped with vectors u, v, and w as
adjacent edges is
V = | u • (v × w) |
.
Example 2: Find the volume of the parallelepiped having
u = 2i +j − 3k, v = i − 2j + 3k, and w = 4i − 3k as
adjacent edges.
The volume is 3 cubic units.
Homework Assignment
Page(s)
Exercises
Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide
Copyright (c) Houghton Mifflin Company. All rights reserved.