Section 7.5 Extra Practice Answers
3. a.
b.
438
5
兹13
⫺9
,
20
4. a. x-axis: 1, (1, 0, 0)
y-axis: ⫺2, (0, ⫺2, 0)
z-axis: ⫺5, (0, 0, ⫺5)
b. x-axis: k, (k, 0, 0)
y-axis: ⫺2k, (0, ⫺2k, 0)
z-axis: ⫺5k, (0, 0, ⫺5k)
5. a. side lengths: 兹5, 兹10, 兹13
angles: 37.87°, 60.26°, 81.87°
b. side lengths: 3, 兹13, 兹14
angles: 48.15°, 63.55°, 68.30°
,a
9
45
, 0,
b
26
兹26 26
b. ␣ ⬟ 115.2⬚, ␤ ⬟ 129.8⬚, ␥ ⬟ 129.8⬚
>
>
>
c. x-axis: OA : 2, OB : 1, OC : 0
>
>
>
y-axis: OA : 2, OB : ⫺1, OC : ⫺1
>
>
>
z-axis: OA : 1, OB : 3, OC : ⫺2
6. a. ⫺
9
7. Answers may vary. For example: (⫺2, ⫺3, ⫺3). In
general, if a vector in R 3 has its tail at the origin and
head with some negative coordinates, this vector will
not usually have direction angles summing to 180°.
,a
兹41
,a
40 60 80
, , b
兹29 29 29 29
8
12 8 4
, a⫺ , ⫺ , ⫺ b
d. ⫺
7
7 7
兹14
c.
15 10
, b
13 13
⫺36 ⫺45
a
,
b
41 41
Calculus and Vectors: Section 7.5 Extra Practice Answers
Copyright © 2009 by Nelson Education Ltd.
>
>
>
1. a. For a on b , extend b into a line through the origin,
and think of this line as a number line, with 0 at
the >origin, positive numbers in the same direction
as b , and negative> numbers in the direction
>
opposite
that of b . Then the scalar projection of a
>
on b is the number you get on this number line if a
>
perpendicular is drawn from the head of a to this
>
line. So a negative scalar projection means that
> a
points in a direction somewhat
> away from b (that
>
is, the angle between a and b is larger than 90°), a
>
positive scalar projection means that
> a points in a
direction somewhat the
> same as b (that is, the
>
angle between a and b is less than 90°),
> and a zero
>
scalar projection means that a and b are
>
perpendicular
to each other (the angle between a
>
and b is exactly
Similar reasoning applies in
> 90°).
>
the case of b on a .
b. Using reasoning similar
> to part a., the vector
>
projection of a on b is not only the number
> you
get on this number line determined by b after
>
projecting a , but this number thought of as a vector
with tail at the origin and head at this number.
>
>
Similar reasoning applies in the case of b on a .
>
2. a. ⫺ i : 1, (⫺1, 0, 0)
>
⫺ j : 1, (0, ⫺1, 0)
>
⫺ k : 1, (0, 0, ⫺1)
>
b. ⫺a i : ⫺1, (⫺1, 0, 0)
>
⫺a j : ⫺1, (0, ⫺1, 0)
>
⫺ak : ⫺1, (0, 0, ⫺1)