Find the point in which the line x=1-t, y=3t, z=1+t; meet the plane 2x-y+3z=6
x = 1 + 2t
y = 2 - 3t
z = -5 + t
Also, we know that:
2x + 5y - 3z = 6
Substituting for x, y and z in the that equation gives:
2 * (1 + 2t ) + 5 * ( 2 - 3t ) - 3 * ( -5 + t ) = 6
2 + 4t + 10 - 15t + 15 - 3t = 6
4t - 15t - 3t = 6 - 2 - 10 - 15
- 14t = - 21 Divide both sides with -14
t = - 21 / -14
t = ( 7 * 3 ) / ( 7 * 2 )
t = 3 / 2
x = 1 - t
x = 2 / 2 - 3 / 2
x = - 1 / 2
y = 3 t
y = 3 * 3 / 2
y = 9 / 2
x = 1 + 2t
y = 2 - 3t
z = -5 + t
Also, we know that:
2x + 5y - 3z = 6
Substituting for x, y and z in the that equation gives:
2 * (1 + 2t ) + 5 * ( 2 - 3t ) - 3 * ( -5 + t ) = 6
2 + 4t + 10 - 15t + 15 - 3t = 6
4t - 15t - 3t = 6 - 2 - 10 - 15
- 14t = - 21 Divide both sides with -14
t = - 21 / -14
t = ( 7 * 3 ) / ( 7 * 2 )
t = 3 / 2
x = 1 - t
x = 2 / 2 - 3 / 2
x = - 1 / 2
y = 3 t
y = 3 * 3 / 2
y = 9 / 2
z = 1 + t
z = 2 / 2 + 3 / 2
z = 5 / 2
Coordinate of point:
( - 1 / 2 , 9 / 2 , 5 / 2 )
Well, look at you asking such fancy geometry questions! It seems like you want to find where this line and plane have a little tete-a-tete. Let's see if we can find their meeting point.
To find the point of intersection, we can simply substitute the values of x, y, and z from the line equations into the plane equation. Let's do some math magic!
For the line:
x = 1 - t
y = 3t
z = 1 + t
Substituting these values into the plane equation:
2(1 - t) - (3t) + 3(1 + t) = 6
Let's simplify:
2 - 2t - 3t + 3 + 3t = 6
Combining like terms:
2 + t = 6
Subtracting 2 from both sides:
t = 4
Now we know the value of t, let's substitute it back into the line equations to find x, y, and z.
x = 1 - 4
x = -3
y = 3(4)
y = 12
z = 1 + 4
z = 5
So, the point where the line intersects the plane is (-3, 12, 5). And there you have it, the meeting point of the line and the plane! Keep those geometry questions coming!
To determine the point at which the line intersects the plane, we need to find the values of t that satisfy both the line and the plane equations.
1. Substitute the line equations into the plane equation:
2(1-t) - (3t) + 3(1+t) = 6
2 - 2t - 3t + 3 + 3t = 6
-2t + 2 + 3t + 3 = 6
t + 5 = 6
t = 1
2. Substitute the value of t back into the line equations to find the point:
x = 1 - t = 1 - 1 = 0
y = 3t = 3 * 1 = 3
z = 1 + t = 1 + 1 = 2
Therefore, the point at which the line intersects the plane is (0, 3, 2).
To find the point where the line and plane intersect, we need to find the values of t that satisfy both the line equations and the equation of the plane. We can then substitute these values of t into the line equations to get the corresponding coordinates.
Step 1:
Let's start by substituting the line equations into the equation of the plane:
2(1-t) - (3t) + 3(1+t) = 6
Simplifying:
2 - 2t - 3t + 3 + 3t = 6
2 - 2t + 3t + 3 + 3t = 6
5t + 5 = 6
5t = 1
t = 1/5
Step 2:
Now that we have found the value of t, we can substitute it back into the line equations to find the coordinates:
x = 1 - (1/5) = 4/5
y = 3(1/5) = 3/5
z = 1 + (1/5) = 6/5
Therefore, the point of intersection between the line and the plane is (4/5, 3/5, 6/5).