Find the projection of the line 3x – y +2z – 1 = 0, x +2y – z = 2 on the plane 3x + 2y +z = 0.
To find the projection of a line onto a plane, we need to find the point of intersection between the line and the plane. Here's how we can find the point of intersection:
Step 1: Write the equations of the line and the plane:
Line: 3x – y + 2z – 1 = 0 ......... (1)
x + 2y – z = 2 ...................... (2)
Plane: 3x + 2y + z = 0 ............ (3)
Step 2: Solve the system of equations formed by equations (1) and (2) to find the point of intersection. We can use the method of substitution or elimination to solve the system. Let's use the substitution method:
From equation (2), solve for x:
x = 2 - 2y + z
Substitute this value of x in equation (1):
3(2 - 2y + z) - y + 2z - 1 = 0
Simplifying, we get:
6 - 6y + 3z - y + 2z - 1 = 0
-7y + 5z + 5 = 0
Rearranging, we have:
-7y + 5z = -5 .............. (4)
Now we have a system of two equations (3 and 4) with two variables (y and z). We can solve this system to find the values of y and z.
Step 3: Solve the system of equations formed by equations (3) and (4):
Multiply equation (4) by 2 to make the coefficient of y in both equations equal:
-14y + 10z = -10 ...........(5)
Now, add equation (3) and equation (5):
3x + 2y + z + (-14y + 10z) = 0 + (-10)
3x - 12y + 11z = -10 ...........(6)
We can see that equation (6) is the equation of a new plane. The point of intersection between the line and the plane is the solution to the system of equations (3) and (4), which is also a point on the new plane.
Step 4: Solve the system of equations formed by equations (3) and (4) to find the values of y and z:
By solving equations (3) and (4), we get:
y = 5/9
z = 10/9
Step 5: Substitute the values of y and z back into equation (2) to find the value of x:
x + 2(5/9) - (10/9) = 2
x + 10/9 - 10/9 = 2
x = 2
So, the point of intersection is (x, y, z) = (2, 5/9, 10/9).
Therefore, the projection of the line 3x – y + 2z – 1 = 0, x + 2y – z = 2 on the plane 3x + 2y + z = 0 is the point (2, 5/9, 10/9).
The line of intersection is
x/-3 = (y+1)/5 = z/7
Now project that line. A nice article is at
http://www.nabla.hr/CG-LinesPlanesIn3DC2.htm