# A Variety of Vector Problems - (pages 37-41)

```Page 37
•
A variety of Vector Problems
1.
Recall the triangle inequality for real numbers says:
for any two real numbers a and b. There is a vector version that says:
for any two vectors a and b. Verify this inequality holds for the specific vectors
u = (-3,2), v = (1, -2). Also verify it for the vectors u = (2, 1) and v = (5, 4).
•
•
3
2.
Find the two wit nonnal vectors to the graph ofthe cwve y = x at the point
(1, 1). Repeat for the point (2, 8).
3.
Forces with magnitudes of 500 pounds and 200 pounds act on a machine part at
angles of30 degrees and -45 degrees with the positive x-axis. Find the direction and
magnitude ofthe resultant forces.
4.
Three forces with magnitudes of 75 pounds, 100 pounds and 125 pounds aU act on the
same object at angles of 30 degrees, 45 degrees and 120 degrees with the positive x-axis.
Find the direction and magnitude ofthe resultant force .
5.
Use vectors to find the points of trisection ofthe line segment with endpoints
(1,2) and (7,5).
6.
Using vectors, prove that the line segment joining the midpoints oftwo sides ofa triangle
is parallel to the third side and has one halfthe length ofthe third side.
7.
Detennine whether the triangle in space with vertices (0, 0, 0), (2,2, 1), (2, -4, 4) is a
right triangle.
8.
Find an equation for the sphere that has center at (-2, 1, 1) and is tangent to the xy-plane.
9.
Find the center and radius of the sphere given by:
10.
use vectors to determine whether the points he in a straight line:
(0, -2, -5), (3,4, 4), (2,2, 1).
II.
find a vector with magnitude exactly 74 and in the same direction a~ the vector
v = (-4,6,2).
12.
Use vectors to find the point that lies exactIytwo-thirds ofthe way from P to Q, where
P=(4, 3,0) and Q == (1, -3, 3).
.~~
Page 38
A VAt'\:t1 of
J)
L
J
-= II (-3)~ II ~ V(-3)''-+lL ==
-= ;11 (I) -~ 1\ -== J,L4(-i)2- -=- IS
/I V'II
cI&lt;&quot;c)
IJ~I\
&lt;:-
U~II
~
:J..
f/3
{u.- +~ jivtrll. VDeWvL
5.f3l{- - &shy;
-YDIA ~o v,~~
.Both
Prolas- so\rd\~
H(-~)~~ +- (t) -~ 1\ = 1l(-a o) Il = J(-J./+o =-
TL +- ~ J( -==
lilt If
&yen; iP - (:\) I) tI.~J V = (5 '1-)
1
fa r4',P-l\dt ClILlo-/' +0 +h~ -6&quot;,
.ho.v~I!~~;L eve
l.A. T\\i-
ti::.( I) I). .'ML. 4t (I) I)
..
V&pound;e-l'O&quot;,
~1t
i4., Y / (I) -:-~x'LI
y
_l!.J_&plusmn;-_ -:-.. :- 3 . Both
tv:3'&quot;,.Ve ~(1.(JOtJ
~
~
J
Vi
-==-: 3L - j
+k ~-,I'.
/~~
=-,~
()\/lc{
ae {~:D
-
-:=:&quot;li -== -3i j
i -
().re-
kj
.
-~)..
3 . -t- - I '
J .
(1
V'11 '--,'
o..v:~d
J//D
'ITO
F = ~&pound;&quot;&middot;k l(S') lL
F ~. ~ ~ : :.
~ Y\ ~ == .' (/ l. (;
't'
~
I so
.,.lA-J'l~+ .. y\ov.W\J.L
cb~~A_=-lZl&middot; = &quot;H~n . ~. /~:~ .. = .V~li
I
3
Ih, e
'&quot;&quot; ., LJ . U
,
&deg;U &middot; I'll. 'I ;.-/'1til)
i91~. ~ i+ IPfC.6 J
f
+
:J..p-v S;&quot;. (- 'IS
(
-;; &cent;
~ /0:1 0
h = 7 SCVnaOj t
Ft- . 70171;
t -= -62.5;
----&quot;7
....&quot;.
FI +
j- -
+~&quot;'1
&cent;=
Le+- ~ b~
.LNv.'lj:hW\
Ft +
75&lt;;.... 30Uj
t 70.7/
~ 3 7. ~j
..
j
1 IPrr.~s-)
=::-!J
j
=-
~ I &quot;, \/ b
7 3. ({P
H&quot;,-
. 6t(. qs;&quot;
Page 39
,
13.16) -rdJ6. Lfh{)
:) p ~
1/.3
~osthd'vt.. V&quot; J...v-
ot-
o
+h&quot;,-
f~,A ~ ,~ +h.L
A ~.(b __ .0. .t'\d&middot;K t~ fp~&middot;rhlfll.. V.t'.~._(Jt f1u- f~)t'\-+ ~-
oL&amp; wa'J ~ kfo B.
. .
•
PI~ -=- off +- _I ~ -== UJ~) + ~ [(lIS) - (t'2J] = 0/) -I- &yen;43)
3
r:
=
ot + ; AB
s=--&plusmn;~ P;\~ ( 3, 3)
-= (}1~ + ~) l) :::. (3) 3)
= 0/1.) + (if Ji)
00&quot;
J
~ (5; if)
(5i L/)--I r j S&quot;C.{- +I-a- I..,,~ .
•
(()
G- \v-l Y\
f)
••
~~
.
O
~ (j( +~
1..
J.~
I
d ~
6c.. -::.
C~
1J
a
0
=
.?,OA- +-
zsc
~ ] -= ~ cJ +- ~ [DC -
J~
J
1- _
'I
-
&quot;
-=
~
.. fr:&middot; (2;;;.,)/.
2'\... +- 1.f'L-f Y'L- ~
08 1.., -==
D}r
-=
{.\ -v\
= DD
.. . . 0&middot;&middot;... ...(~)~~~.;
_
-
P&shy;
- ~ 0&laquo; =
~ T J 02 - 'Aofl- /-.~~
~ [o~ - oA]
d-k
~
~
[~
Show: ~K
g
1\
II
r:L~ + ~ ot
I
,
0
B
~It- ==Ci
&quot;
Page 40
'L
+ 2.L+ It... -=-
.--.----.. -- .
8-&middot;-&middot;Ti.,::'v;~)&middot;&middot;---_&middot;_--
3&amp;
a
.{
M~ = o,-+&amp;'&quot;L+ ,)'1.- ~ l/5
.' .
b.
0
IrB
IS
~pk-hj 'Ov.&quot;,~
50
~
r&middot;~ht ~ &quot;jk~J
0...
j
ex -&plusmn;) l.t ~+ ~ '-.-t- t&quot;L
1~14
-= Cr.. 1- )1&raquo;
1
+k. L}..W&gt;-:k. 4-
)
~. I,
et~~UI1H''''--
+~~ (JV~. wt .
:=.. (
&reg;
Ii ~ (0 J -
g
c8
&quot;,
2, - ,) /
-== (3/ hi q)
=-(~ )/3)
C~
/1
J;0
f
J &lt;.)
8 ~ ( 3/ ~I()} c
~
-: -
=- ('1./ 2; 0
Page 41
..
3 cB
0_. a.lr . 3 t~ll1-k . ck4);1J~ .. faJ71/~(0c1ovJL J
~.cu_-j-4j J:-....... .f1..- SAM.&lt;-
l&quot;&quot;, ,
3 (-'l.(,~,'J..)
.&quot;C(
J ~&quot;--rtll-+~i.
-:::
t 3 ) Cf/tj 3/ y
J z:;&amp;
Lt-e&shy;
~
+
3[O~-
Of) =&shy;
C'i/)/~ + ~ [(-~J -b/3)j =
A.. _~_ . . '&plusmn;k P01.d:- ( 9) -I} J)
•
&quot;
•
```