Page 42
More vector Problems - dot product
1.
For the following vectors u and v, find the following quantities:
(a) u' V
(b) v' v
(c) jluW
(d) (UI v)v
(e) u' (2v)
(i) u = (1. -3. 5) and v = (-2, -3, 4)
(ii) u = (1, 0, -2) and v = (0.3, -5)
2.
Find the inner product (dot product) ofu and v if the magnitudes ofu and v are,
respectively, 10 and 6 and the angle acute angle between them is 60 degrees.
3.
Find the angle between the vectors u = 3i - 2j + 5k and v = 5i -3j+7k.
4.
Determine whether u and v are orthogonal, parallel or neither:
u = (4,3) and v = (112, -2/3).
5.
Find the angle between a cube's main diagonal and one of its edges. Also find the
angle between a cubes main diagonal and the diagonal of its square base.
6.
Find two vectors in opposite directions that are orthogonal to v = -8i+3j.
7.
Find the work done in moving an object from P to Q ifthe magnitude and
direction ofthe force are given by the vector v:
P(O, 0, 0), Q(4, 7, 5), v= 0,4,8).
8**.
Prove the Caucby-Scbwarz Inequality:
[Hint: This inequality requires a rather clever idea. Consider the real function
/(1)
=II III + l'1I2 ,
for all real numbers t
Note that f(t) is never less than zero and can also be expressed using the dot
product - do that and then apply basic properties of the dot product listed in your
text in order to expand the resulting expression...now consider that expression as a
quadratic function (parabola) in the variable 1.... now its up to you to make the
clever observation to finish the proof1].
.'t\!
CD
DbC
(L) I? -= Q} -3} S-) ) V ~
P,.J" t+
Page 43
(-;)..) 4/f)
SD ,....t;~
- . tA 'V -= '\(-:t) +(-3'(-3) + S( 4) ~ ~rv.1 -= 0-i')'L +- ~_3)L -+ ~L = ~ ~
IlW/I 'L -= It· it = \~ +(_3j'L + Sl- =- 3 S
[1' ~ l' -= ~ (if·v) -= ~(~ i) = 5~
\it . V) \t ~ 'al V -==- (- S"l{) -61)__ _----,-----_
\ Dfi)
tiL) lA -= U) 0) -~)) V = (0) 3} -S}
--'
~------... _._------_._---_._-~------------_
..
....
... __ ._ ..
_---_._. __ .. __ ..__ ... _....__ ..... __... --_
.....
It. V -= \( 0) -+- () ( 3) ~ (- 2) (- ~) -=­ \ 0
V, V -= 0'1- + ~l--+ ~ ~)'L -=- 3 Y
Uktl = lZ· It' = 11.. + 01. +- (-'2..) -= S
\....
'L
. . (II.V)V ~ 10\1' -= (0; '30 I-~
U· ~V =- ~c;trV) -= Jl )0) ~ ).0
riJ-Lt~-\f~-;I_fu/i~I~-~{ID){~)~()6::=y1.---­
Q)
~0
Page 44
CD it· V :::.
II I
.--'--~--
®
U
£',l
"'J
!f(
11.)
V
fA
+
re-
3
DV+-~CJ ~frv\~
-.._ --_._._-_ _---_._._.--_._---_ _..-._ - 11='( IJ~O
4
_.. _ .
1b -f;'~J
-4'
~~~~J
lA'j
A-lo-o
U·j
-=
. - ... - ." ... .. _- .. - - ...-.... . .
. ..
- _.
_. .
..
tl"?:~ ,L.~""'-~~&~~cJ
tl.<J.....:r=A-t~
/
e
x
')(-2)::: 0
-=
I
I·
\~I)I)·(o)l)D)
=I
llUIl ~ j 11 ~ e- -=- J:3
Cr.J9­
:. l-==-J:3~eo
e- ~ 5~(7L{
Lcd~A.1/J. ~ It a ",1.. V ;
It'. if -= OJ I) 1) . (~ ,) 0) = L
LA ,V =:: /I (I) 'Joll HOJ I}D)" ~ ep
9- =- f3 {iJ...· f
o
t/J ~ 3S,Jb
•
.
..
..
•
(f)
~
~
Q
v' P -=-
wo vl -==
(1)'1) <t)
I
(~) 7/;)
=
'~\
l{ + 2 Y+-l( 0
Page 45
12.
••
-=
=
~4
Sw:e 1\ ti(j:\f\\f..
>
0)
(ik/l)-e- + 2{ [{, it) t + 11· V' ?- 0
~
~
------­
ABC­
c+ ~. .
~__nJ;( vo1~
t
v + V, v
~(lf. v) t + V· if
t'L Lt, It + at U'
(it-lK) t'L -I-
-..»~
-1J4wiwQj ~'I:.- ~I\d.jn.- ~- A(-+gc -tC
,I'(j
~. . ±k.. t~n~)+k dj~I",;n",,-l- f3~llA-:C« 0
.
-.(L~'4}'.'Jafr~ +v ~tk. t-V>tk~J ~lt~~""'Q
,ervalifllJilJA.d:; l'\-l C/4AtA.~\~ .~·',LloL 11 E M uh ~ !).- ." .
J,,,,,,,:
...
!f( itVt" -L{(~lf)(V·Yl~C/
. , 4 ( i{.
vy- - ~11U1II1VII1."L
lt· \1) =: lI~i IIV~L
~ ~l
,
J
] lA ·
v
<
­
lJI11l1lvll
C.
0
.