Intersecting Planes & Lines: Point of Intersection

[spoc21]

Homework Statement

Find out if the following planes and lines intersect. If they intersect, state the point of intersection

Plane: 2x + y + 3z = 10
Line: Passing through the point A(1, 5, 1) and B(0, 4, 2)

Homework Equations

The Attempt at a Solution

I have solved the problem, but am unsure if my working/result is correct..

first we have to find the equation for the line:

[x,y,z]=(0,4,2)+t[1,5,1]

Equation of the line:

x=1t
y=4+5t
z=2+t

We have to know check if they intersect:

2x + y + 3z = 10

Substitute the line equation in the line :
2(t)+(4+5t) + 3(2+t) = 10

2t+4+5t+6+3t=10

10t+10=10

10t=0

t=0

The lines and the plane intersect, since t is a number (0).

Find the point of intersection

Substitute the value of t into the parametric equations:

x=1(0)
x = 0

y=4+5(0)
y = 4

z=2+(0)
z = 2


so the point of intersection is (0,4,2)..but this is the point given in the original question..I am very confused over here...any help is much appreciated

Thanks!

 

[Mark44]

Homework Statement

Find out if the following planes and lines intersect. If they intersect, state the point of intersection

Plane: 2x + y + 3z = 10
Line: Passing through the point A(1, 5, 1) and B(0, 4, 2)

Homework Equations

The Attempt at a Solution

I have solved the problem, but am unsure if my working/result is correct..

first we have to find the equation for the line:

[x,y,z]=(0,4,2)+t[1,5,1]

This isn't right. You can't just take the two points and plunk them into your parametric equation. You need to find a vector with the same direction as the line. Use the two given points to do this.

Equation of the line:

x=1t
y=4+5t
z=2+t

We have to know check if they intersect:

2x + y + 3z = 10

Substitute the line equation in the line :
2(t)+(4+5t) + 3(2+t) = 10

2t+4+5t+6+3t=10

10t+10=10

10t=0

t=0

The lines and the plane intersect, since t is a number (0).

Find the point of intersection

Substitute the value of t into the parametric equations:

x=1(0)
x = 0

y=4+5(0)
y = 4

z=2+(0)
z = 2


so the point of intersection is (0,4,2)..but this is the point given in the original question..I am very confused over here...any help is much appreciated

Thanks!