APP. A] VECTORS AND MATRICES 423
SolvedProblems
VECTORS
A.1. Letu=( 2 ,− 7 , 1 ),v=(− 3 , 0 , 4 ), andw=( 0 , 5 ,− 8 ). Find: (a)3u− 4 v;(b)2u+ 3 v− 5 w.
First perform the scalar multiplication and then the vector addition.(a)3u− 4 v= 3 ( 2 ,− 7 , 1 )− 4 (− 3 , 0 , 4 )=( 6 ,− 21 , 3 )+( 12 , 0 ,− 16 )=( 18 ,− 21 ,− 13 ).
(b)2u+ 3 v− 6 w= 2 ( 2 ,− 7 , 1 )+ 3 (− 3 , 0 , 4 )− 5 ( 0 , 5 ,− 8 )=( 4 ,− 14 , 2 )+(− 9 , 0 , 12 )+( 0 ,− 25 , 40 )
=(− 5 ,− 39 , 54 ).A.2. For the vectoru,v,win Problem A.1, find: (a)u·v;(b)u·w;(c)v·w.
Multiply corresponding components and then add:(a) u·v= 2 (− 3 )− 7 ( 0 )+ 1 ( 4 )=− 6 + 0 + 4 =− 2.
(b) u·w= 2 ( 0 )− 7 ( 5 )+ 1 (− 8 )= 0 − 35 − 8 =− 43.
(c) v·w=− 3 ( 0 )+ 0 ( 5 )+ 4 (− 8 )= 0 + 0 − 32 =− 32.A.3. Find‖u‖where: (a)u=( 3 ,− 12 ,− 4 );(b)u=( 2 ,− 3 , 8 ,− 7 ).
First find‖u‖^2 =u·uby squaring the components and adding. Then‖u‖=√
‖u‖^2.(a) ‖u‖^2 =( 3 )^2 +(− 12 )^2 +(− 4 )^2 = 9 + 144 + 16 = 169 .Hence‖u‖=√
169 = 13.
(b) ‖u‖^2 = 4 + 9 + 64 + 49 = 126 .Hence‖u‖=√
126.A.4. Findxandyifx( 1 , 1 )+y( 2 , 1 )=( 1 , 4 ).
First multiply by the scalarsxandyand then add:x( 1 , 1 )+y( 2 ,− 1 )=(x, x)+( 2 y,−y)=(x+ 2 y, x−y)=( 1 , 4 )Two vectors are equal only when their corresponding components are equal; hence set the corresponding components
equal to each other to obtainx+ 2 y=1 andx−y=4. Finally, solve the system of equations to obtainx=3 and
y=−1.A.5. Supposeu=⎡
⎣5
3
− 4⎤
⎦,v=⎡
⎣− 1
5
2⎤
⎦,w=⎡
⎣3
− 1
− 2⎤
⎦. Find: (a)5u− 2 v;(b)− 2 u+ 4 v− 3 w.(a)5u− 2 v= 5⎡
⎣5
3
− 4⎤
⎦− 2⎡
⎣− 1
5
2⎤
⎦=⎡
⎣25
15
− 20⎤
⎦+⎡
⎣2
− 10
− 4⎤
⎦=⎡
⎣27
5
− 24⎤
⎦.(b) − 2 u+ 4 v− 3 w=⎡
⎣− 10
− 6
8⎤
⎦+⎡
⎣− 4
20
8⎤
⎦+⎡
⎣− 9
3
6⎤
⎦=⎡
⎣− 23
17
22⎤
⎦.MATRIX ADDITION AND SCALAR MULTIPLICATION
A.6. Find 2A− 3 B, whereA=
[
1 − 23
45 − 6]
andB=[
302
− 718]
.First perform the scalar multiplications, and then a matrix addition:2 A− 3 B=[
2 − 46
810 − 12]
+[
− 90 − 6
21 − 3 − 24]
=[
− 7 − 40
29 7 − 36](Note that we multiplyBby−3 and then add, rather than multiplyingBby 3 and subtracting. This usually avoids
errors.)