James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Chapter 12
Concept Check Answers
Cut here and keep for reference
1. What is the difference between a vector and a scalar?
A scalar is a real number, whereas a vector is a quantity that
has both a real-valued magnitude and a direction.
2. How do you add two vectors geometrically? How do you add
them algebraically?
To add two vectors geometrically, we can use either the
Triangle Law or the Parallelogram Law:
u+v
u
Triangle Law
v
v
u
v+u
u
u+v
Parallelogram Law
Algebraically, we add the corresponding components of the
vectors.
v
7. Write expressions for the scalar and vector projections of b
onto a. Illustrate with diagrams.
Scalar projection of b onto a: comp a b − a b
| a |
b
comp a b
Vector projection of b onto a:
proj a b −S a b
| a |
D
a
a
| a | − a b
| a | a 2
3. If a is a vector and c is a scalar, how is ca related to a
geo metrically? How do you find ca algebraically?
For c . 0, ca is a vector with the same direction as a and
length c times the length of a. If c , 0, ca points in the direction
opposite to a and has length | c | times the length of a.
Algebraically, to find ca we multiply each component of a
by c.
b
proj a b
a
4. How do you find the vector from one point to another?
The vector from point Asx 1, y 1, z 1d to point Bsx 2, y 2, z 2d is
given by
kx 2 2 x 1, y 2 2 y 1, z 2 2 z 1l
5. How do you find the dot product a b of two vectors if you
know their lengths and the angle between them? What if you
know their components?
If is the angle between the vectors a and b, then
a b − | a | | b | cos
If a − ka 1, a 2, a 3l and b − kb 1, b 2, b 3l, then
a b − a 1b 1 1 a 2b 2 1 a 3b 3
6. How are dot products useful?
The dot product can be used to find the angle between two
vectors. In particular, it can be used to determine whether two
vectors are orthogonal. We can also use the dot product to find
the scalar projection of one vector onto another. Additionally,
if a constant force moves an object, the work done is the dot
product of the force and displacement vectors.
8. How do you find the cross product a 3 b of two vectors if
you know their lengths and the angle between them? What if
you know their components?
If is the angle between a and b (0 < < ), then a 3 b is
the vector with length | a 3 b | − | a | | b | sin and direction
orthogonal to both a and b, as given by the right-hand rule. If
then
a − ka 1, a 2, a 3l and b − kb 1, b 2, b 3l
a 3 b −
Z
i
a 1
b 1
j
a 2
b 2
Z
k
a 3
b 3
− ka 2b 3 2 a 3b 2, a 3b 1 2 a 1b 3, a 1b 2 2 a 2b 1l
9. How are cross products useful?
The cross product can be used to create a vector orthogonal
to two given vectors and it can be used to compute the area
of a parallelogram determined by two vectors. Two nonzero
vectors are parallel if and only if their cross product is 0. In
addition, if a force acts on a rigid body, then the torque vector
is the cross product of the position and force vectors.
(continued)
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