HW07 – gilbert – (55035)
1
This print-out should have 18 questions.
Multiple-choice questions may continue on
the
next
column
or
page – find
all
choices before answering.
1
10.0 points
A
triangle
P QR
in 3-space has vertices
P
(
−
1
,
3
, −
2)
, Q
(3
,
5
, −
3)
, R
(
−
2
,
6
,
0)
.
Use vectors to decide which one of the
follow-ing properties the triangle has.
1. right-angled at Q
2. not right-angled at P, Q,
or R
3. right-angled at P
correct
4. right-angled at R
Explanation:
Vectors
a
and
b
are perpendicular
when
a · b
= 0. Thus
P QR
will be
−−→
−→
(1)
right-angled at
P
when
QP
·
RP
= 0,
−−→
−−→
(2)
right-angled at
Q
when
P Q
·
RQ
= 0,
−→
−−→
(3)
right-angled at
R
when
P R
·
QR
= 0. But for the vertices
P
(
−
1
,
3
, −
2)
, Q
(3
,
5
, −
3)
, R
(
−
2
,
6
,
0)
we see that
keywords: vectors, dot product, right
trian-gle, perpendicular,
002
10.0 points
Find the scalar projection of
b
onto
a
when
b
= 2
i
+ 3
j −
2
k ,
a
= 2
i −
j −
2
k .
1.
scalar projection
=
2
2.
scalar projection
=
4
3
3.
scalar projection
=
1
4.
scalar projection
=
5
3
correct
5.
scalar projection
=
7
3
Explanation:
The scalar projection of
b
onto
a
is
given in terms of the dot product by
comp
A
b
=
a
·
b
.
|a|
Now when
b
= 2
i
+ 3
j −
2
k ,
a
= 2
i −
j −
2
k ,
we see that
−−→
PQ
=
4
,
2
,
1
,
−−→
QR
=
5
,
1
,
3
,
h
−
O
h −
O
while
−→
RP
=
1
,
3
,
2
.
h
−
−
O
Thus
−−→
QP
·
−→
RP
=
0
,
−−→
PQ
·
−−→
RQ
=
21
,
and
−→
PR
·
−−→
QR
=
−
14
.
Consequently,
P QR
is
right-angled at P
.
a·b
= 5
,
|a|
=
q
(2)
2
+ (
−
1)
2
+ (
−
2)
2
.
Consequently,
comp
A
b
=
5
3
.
keywords:
003
10.0 points
The box shown in