5c-12.4-5.math

12.4 35
P(-2,1,0) Q(2,2,2) R(1,4,-1) S(3,6,4)

PQ = <2-(-2), 2-1, 2-0> = <4, 1, 2>
PR =<1-(-2), 4-1, -1-0> = <3, 3, -1>
PS =<3-(-2), 6-1, 4-0>  = <5, 5, 4>

Triple Product
     30 -20 12
4 1 2  4 1
3 3 -1 3 3
5 5 4  5 5
     48 -5 30
     
73-(22) = 51
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12.4 37

u = i + 5j − 3k, v = 2i − j, and w = 5i + 9j − 6k

      15 0 -60
1 5  -3 1 5
2 -1 0  2 -1
5 9  -6 5 9
      6  0  -27(2)

-40-14+6-(15-60)
-54+6+45
-54+51=-3

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12.4 11

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12.5 13

Is the line through 
(−4, −6, 1) and (−2, 0, −3)
 parallel to the line through 
(10, 18, 7)
 and 
(5, 3, 17)?

<-2--4, 0--6,-3-1> = <2,6,-4>
<5-10,3-18,17-7> = <-5,-15,10>

|<2,6,-4>|=sqrt(4+36+16)=sqrt(56)=2sqrt(14)
|<-5,-15,10>|=sqrt(25+225+100)=sqrt(350)=5sqrt(14)


<2/2sqrt(14), 6/2sqrt(14), -4/2sqrt(14)>
= <1/sqrt(14),3/sqrt(14),-2/sqrt(14)>

<-5/5sqrt14, -15/5sqrt14,10/5sqrt(14)>
= <-1/sqrt14, -3/sqrt14, 2/sqrt(14)>

yes
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12.5 33
Find an equation of the plane.
The plane through the points 
A(2, 1, 2),
 
B(3, −8, 6),
 and 
C(−2, −3, 1)

AB = <3-2,-8-1,6-2>
   = <1,-9,4>
AC = <-2-2,-3-1,1-2>
   = <-4,-4,-1>

Normal Vector

        36k -16i -j
 i  j  k  i  j 
 1 -9  4  1 -9
-4 -4 -1 -4 -4
       9i -16j -4k

9i-16j-4k-(36k -16i -j)
= 9i-16j-4k-36k+16i+j
= 25i-15j-40k

=> 25(x-2)-15(y-1)-40(z-2)=0
=> 25x-50-15y+15-40z+80=0
=> 25x -15y -40z +45=0

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12.5 26

Find an equation of the plane.
The plane through the point 
(4, 0, 8)
 and perpendicular to the line 
x = 4t,
 
y = 8 − t,
 
z = 7 + 3t

=> 4(x-4)-1(y-0)+3(z-8)=0
=> 4x-16-y+3z-24=0
=> 4x-y+3z-40=0

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12.5 27

Find an equation of the plane.
The plane through the point 
(7, −7, −8)
 and parallel to the plane 5x − y − z = 6
 
=> 5(x-7)-(y+7)-(z+8)=0
=> 5x-35-y-7-z-8=0
=> 5x-y-z-50=0
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12.5 11
Find parametric equations for the line. (Use the parameter t.)
The line through 
(−8, 2, 5)
 and parallel to the line 
x/2 = y/3 = z + 1

(x+8)/2 =(y-2)/3=(z-5)/1

x=2t-8
y=3t+2
z=t+5

(2t-8,3t+2,t+5)
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12.5 19
Determine whether the lines 
L1
 and 
L2
 are parallel, skew, or intersecting.
L1: x = 6 + 4t,  y = 8 − 2t,  z = 2 + 6t
L2: x = 3 + 12s,  y = 9 − 6s,  z = 12 + 15s

(x-3)/12=(y-9)/-6=(z-12)/15

(6+4t-3)/12=(8-2t-9)/-6=(2+6t-12)/15
[(3+4t)/12=(-1-2t)/-6=(-10+6t)/15]60
5(3+4t)=-10(-1-2t)=4(-10+6t)
15+20t=10+20t=-40+24t
40+15+20t=40+10+20t=24t
55+20t=50+20t=24t
=>false