LINES AND PLANES: needs checking

[livestrong136]

NEED CORRECTION, also this . means dot multiplication.

My teacher comments:

#6) you've made some errors (-2 marks)

#8) correct, they intersect at a point, but you need to find the point like you did in #7 for full marks (-3 marks)


6.Determine the intersection, if any, of the planes with equations x + y – z + 12 =0 and 2x + 4y - 3z + 8 = 0.

The normal vectors for the two planes are (1, 1, -1) and (2, 4, -3).
- These vectors are not collinear therefore the planes intersect in a line.
x+y-z = -12 (1)
2x+4y-3z = -8(2)

-3(1) + 2: -x + y = 28 = x+28
Let x = t.
y = t+28
Substituting in (1)
One of the either answers=>
t+t+28-z = -12 or z = 2t+40.
The parametric equations for the line of intersection are
x = t, y = 28+t, z = 40+2t.

8. Give a geometrical interpretation of the intersection of the planes with equations
x + y − 3 = 0
y + z + 5 = 0
x + z + 2 = 0
N1= (1, 1, 0) N2= (0, 1, 1), N3= (1, 0, 1)
N1 x N2
= ((1,1,0) x (0,1,1)) . (1,0,1)
= (1,-1,1) . (1,0,1)
=2
(N1 x N2) . N3 ≠ 0
Since the triple dot product does not equal to 0, then these three planes must intersect in a single point.

 

[Mark44]

NEED CORRECTION, also this . means dot multiplication.

My teacher comments:

#6) you've made some errors (-2 marks)

#8) correct, they intersect at a point, but you need to find the point like you did in #7 for full marks (-3 marks)


6.Determine the intersection, if any, of the planes with equations x + y – z + 12 =0 and 2x + 4y - 3z + 8 = 0.

The normal vectors for the two planes are (1, 1, -1) and (2, 4, -3).
- These vectors are not collinear therefore the planes intersect in a line.
x+y-z = -12 (1)
2x+4y-3z = -8(2)

-3(1) + 2: -x + y = 28 = x+28
Let x = t.
y = t+28
Substituting in (1)
One of the either answers=>
t+t+28-z = -12 or z = 2t+40.
The parametric equations for the line of intersection are
x = t, y = 28+t, z = 40+2t.

8. Give a geometrical interpretation of the intersection of the planes with equations
x + y − 3 = 0
y + z + 5 = 0
x + z + 2 = 0
N1= (1, 1, 0) N2= (0, 1, 1), N3= (1, 0, 1)
N1 x N2
= ((1,1,0) x (0,1,1)) . (1,0,1)
= (1,-1,1) . (1,0,1)
=2
(N1 x N2) . N3 ≠ 0
Since the triple dot product does not equal to 0, then these three planes must intersect in a single point.

For #8 you have a system of 3 equations in 3 variables. You can solve the equations using algebraic techniques, or you can write the equations as an augmented matrix, and then row reduce the matrix.

 

[livestrong136]

? I am confused

 

[Mark44]

? I am confused

Saying that you are confused doesn't help. What are you confused about?