Study Guide
B l o c k 1: V e c t o r A r i t h m e t i c
U n i t 4: The D o t P r o d u c t
1.
L e c t u r e 1.040
* S e e n o t e o n n e x t page. 1.4.1
Study Guide
Block 1: Vector Arithmetic
Unit 4: The Dot Product
Lecture 1.040 continued
Note:
L a s t equation on f i r s t board (a) should read:
Study Guide
Block 1: Vector A r i t h m e t i c
U n i t 4: The Dot Product
2.
Head Thomas 12.6.
3.
Exercises :
a.
+
+
+
+
+
k and l e t B = 2 i + 3 j + 4k.
Find t h e r e l a t i o n s h i p
+ +
+ +
+
++
+
between x, y , and z i f A - B = A - C , where C = x l + y j + zk.
2.
+
+
Let A = i
+
+
j
+
U s e v e c t o r a r i t h m e t i c t o g e n e r a l i z e t h e r e s u l t of p a r t ( a ) by
+
+ +
+ +
+
+
showing t h a t i f A - B = A * C and w e know t h a t
#
and B # C , t h e n A
+
-
2
+
6
c.
i s perpendicular t o B
C..
-+
+
With A and B a s i n p a r t ( a ) ,. u s e t h e r e s u l t of p a r t ( b ) t o i n t e r +
+ +
+ +
p r e t g e o m e t r i c a l l y t h e set of a l l v e c t o r s C f o r which A n B = A'C.
d.
I n t e r m s of s t r u c t u r e , e x p l a i n why t h e r e s u l t of t h i s e x e r c i s e i s
n o t a c o n t r a d i , c t i o n of t h e c a n c e l l a t i o n l a w of n u m e r i c a l a r i t h m e t i c .
a.
+
+
L e t P 0 ( 2 , 3 , 4 ) be i n t h e p l a n e M and l e t V = 5 i
p e n d i c u l a r t o M.
+
+
8j
+
+
6k be p e r -
Determine t h e r e l a t i o n s h i p between x, y , and z
i f P ( x , y , z ) i s t o l i e i n t h e p l a n e M.
(This r e l a t i o n i s c a l l e d
t h e , e q u a t i o n of t h e p l a n e i n C a r t e s i a n c o o r d i n a t e s . )
b.
Is ( 1 , 1 , 8 ) i n t h e p l a n e M?
plane?
I f n o t , where i s i t . r e l a t i v e t o t h e
Explain.
Lines a r e drawn from A ( 1 , 2 , 3 ) t o b o t h B ( 3 , 3 , 5 ) and C ( 4 , 4 , 9 ) .
+
-+
Determine t h e measure of 3 BAC ( i e , t h e a n g l e between AB and AC)
..
1.4.4 (L)
F i n d t h e p o i n t a t which t h e l i n e through A ( 2 , 3 , 4 ) p e r p e n d i c u l a r t o
t h e l i n e y = -3x + 2 m e e t s t h e l i n e y = -3x + 2.
1.4.5(L)
+
+
+
+
+
+
L e t v, = 3 i + 2 j + 5k and l e t v, = 2 i
+
+
-+
+&
v a l u e of (3vl + 4v2)
(4vl
3v2).
J.
-
+
-+
7j
+
+
3k.
Determine t h e
.
1.4.8(L)
Use v e c t o r methods to show t h a t t h e perpendicular d i s t a n c e from
t h e p o i n t A ( x o , y o ) t o t h e l i n e ax
+
by
+
c = 0 i s g i v e n by
The l i n e L i s determined by t h e two p o i n t s i n s p a c e ,
(2,4,5).
t o L.
( 1 , 2 , 3 ) and
Find t h e p e r p e n d i c u l a r d i s t a n c e from t h e p o i n t (3,5,9)
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