12.3

1) dot <1,2,1> <4,3,5>
=(1*4)+(2*3)+(1*5)=4+6+5=15

13) Orthogonal? <1,1,1> <1,-2,-2>
=(1*1)+(1+-2)+(1*-2)=1+-2+-2=1-4=-3
-3 not equal to 0 so not orthogonal
since -3<0 it's obtuse

21) Cos? <i+j> <j+2k>
=(i*j)+(j*2k)/(sqrt(i^2+j^2)*sqrt(j^2+2k^2))
=(1+2)/sqrt(2)*sqrt(5)=
*****

29) Orthogonal? <b,3,2> <1,b,1> and <4,-2,7> <b^2,b,0>
(b+3b+2=0)=(4b=-2)=(b=-2/4)=(b=-1/2)
(4b^2-2b=0)=(b=0 or b=1/2)

31) Orthogonal? <2,0,-3>
(2*a)+(0*b)+(-3*c)=0--> <0,5,-1/3> and <1/2,2,0>

57) Projection u=<5i+7j-4k> v=<k>
((0+0+(-4*1))/(1^2))*1=(-4)/1=-4k


63) OP? <8,2>
sqrt(8^2+2^2)=2*sqrt(17)
*****

12.4

1) <1,2> X <4,3>
=(1*3-2*4)=3-8=-5

5)  <1,2,1>(<4,-3,0> X <1,0,1>)
=1(-3*1-0*0)-2(4*1-0*1)+1(4*0--3*1)
=-3-8+3=-11+3=-8

13) (i+j) X k
= iXk + jXk= -j+i

21) (u-2v) X (u+2v)
(u-2v)=
*****

25) u
*****

27) <3,0,0>X<0,1,-1>
i j k
3 0 0 ---> i(0)-j(-3)+k(3)
0 1 -1

39) Volume? <1,0,0> <0,2,0> <1,1,2>
|1,0,0.((0,2,0)X(1,1,2))|
i j k
0 2 0 ---> i(4)-j(0)+k(-2)=<4-0-2>.<1,0,0>=4+0+0=4
1 1 2

41) Area? <1,0,3> X <2,1,1>
i j k
1 0 3 ---> i(-3)-j(1-6)+k(1)=-3+5+1=sqrt(3^2+5^2+1^2)=sqrt(35)
2 1 1

43) Area? <3,3,0> X <0,3,3>
i j k
3 3 0 ---> i(9)-j(9)+k(9)=9-9+9=9 sqrt(9^2+9^2=9^2)=(9*sqrt(3))/2
0 3 3

45) P(1,2) Q(3,4) R(-2,2)
PQ=(2,2) PR=(-3,0)
 2 2
-3 0
=0--6=6=sqrt(6^2)/2=3

12.5

1) n=<1,3,2> <4,-1,1>
1(x-4)+3(y--1)+2(z-1)=0
x-4+3y+3+2z-2=0
x+3y+2y=3

5) n=<i> <3,1,-9>
i(x-3)=0
ix-3i=0 
x=3

9)
x=0

11) True for plane parallel to yz-plane
n=(1,0,0) is normal vector
equation has form x=d

13) 9x-4y-11z=2
<9,-4,-11>

15) 3(x-4)-8(y-1)+11z=0
3x-12-8y+8+11z=0
3x-8y+11z=4
<3,-8,11>

17) P=(2,-1,4) Q=(1,1,1) R=(3,1,-2)
PQ=(-1,2,-3) PR=(1,2,-6)
 i j  k
-1 2 -3 --->i(-6)-j(9)+k(-4)
 1 2 -6
2(x+6)-1(y+9)+4(z+4)=0
2x+12-y-9+4z+16=0
-6x-9y-4z=-19

19) P=(1,0,0) Q=(0,1,1) R=(2,0,1)
PQ=(-1,1,1) PR=(1,0,1)
 i j k
-1 1 1 ---> i(1)-j(2)+k(-1)
 1 0 1
2(x-1)+0(y+2)+1(z+1)=0
2x-2+z+1=0
2x+z=1
*****

25) (-2,-3,5) normal=i+k=<1,0,1>
1(x+2)+0+1(z-5)=0
x+2+z-5=0
x+z=3

31) x=4 y=4 z=4

53) R^3 xz-plane 3x+2z=5
(3%)x+by+(2%)z=5%


13.1

5) P=(3,-5,7) v=<3,0,1>
<3,-5,7>+(t)<3,0,1>
=<3,-5,7>+<3t,0,t>=<3+3t,-5,7+t>

7) r(t)=<t^2,t^2-2t-3,t-3>
yea

13.2

3) i+4k

5) <-t^-2,cost,0>

7) <1,2t,3t^2>

15) no

31) <1-t^2,5t,2t^3> t=2 --> <-3,10,16>
<-2t,5,6t^2> t=2
<-4,5,24>
<-3,10,16>+(t)<-4,5,24>
<-3,10,16>+<-4t,5t,24t>
<-3+-4t,10+5t,16+24t>

33) <4s^-1,0,-8/3s^-3> s=2 --> <2,-1/3>
<-4s^-2,0,8s^-4> s=2 --> <-1,1/2>
<2,0,-1/3>+(t)<-1,0,1/2>
<2,0,-1/3>+<-t,0,t/2>
<2-t,0,-1/3+t/2>


41) u^4/4+u^6/6|2 to -2
=(0,0)

49) t^3/3+5t^2/2+t+C t=1
<0,1,2>=<1/3,5/2+1+C>
C=<-1/3,-3/2,1>
<(t^3/3)-1/3,(5t^2/2)-3/2,t-1>