Randomz96
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Hi people, I have to find the line of intersection of 2 planes. Plane 1 equ: r.(3i-2j-k)=5 and plane 2 equ: r.(4i-1j-2k)=5. The line of intersection needs to have the form r= a+ub

Does anyone know how to find any point on the line common to both planes (i.e. a), I already have the gradient (b) but I need a to write the equation of the line of intersection. The gradient of the line I calculated to be (3i2j5k)

thanks ������
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Smaug123
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(Original post by Randomz96)
Hi people, I have to do with the line of intersection of 2 planes. Plane 1 equ: r.(3i-2j-k)=5 and plane 2 equ: r.(4i-1j-2k)=5. The line of intersection needs to have the form r= a+ub

Does anyone know how to find any point on the line common to both planes (i.e. a), I already have the gradient (b) but I need a to write the equation of the line of intersection. The gradient of the line I calculated to be (3i2j5k)

thanks ������
Your gradient is quite correct.

How can you find a point on both lines? Well, suppose such a point is {x,y,z}. It's very likely (unless your line is accidentally really unlucky) that the line will at some point pass through the plane x=0. Therefore, you can assume that a point on the line is {0,y,z}, and try to find such a point. (In the very unlikely case that that fails, repeat with y=0 instead.)

Can you find a point {0,y,z} such that {0,y,z}.{3,-2,-1} = 5 and {0,y,z}.{4,-1,-2} = 5?
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VannR
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Full general technique for solving this type of problem that I put down during last year's revision:

1. If it has not been done already, express the equations of the planes in the form r.n = p

2. Find the cross product of the normals of the two planes. This vector will be perpendicular to both normals, and is therefore in the direction of the line of intersection of the two planes.

3. We now need to find a point the lies on the line. Express the planes in Cartesian form. Substitute a value for one of your variables, such as x = 0, into both equations. Solve the equations for the other two variables.

4. A point on the line of intersection and its direction is now known. Substitute this information into the equation for a straight line.
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