The Point of Intersection of a Line and a Plane

Problem: Find the point of intersection of the line having the position vector equation r1 = [2, 1, 1] + t[0, 1, 2] with the plane having the vector equation r2.[1, 1, 2] = 3
A diagram of this is shown on the right.		O is the origin. P is the point of intersection of the line and the plane. is the position vector of any point on the line. is the position vector of any point in the plane.
The position vectors are shown drawn at the point of intersection of the line and the plane where = .		Since, at the point of intersection, the two position vectors are identical it follows that; we can substitute = [2, 1, 1] + t[0, 1, 2] for in; .[1, 1, 2] = 3 that is; ([2, 1, 1] + t[0, 1, 2]) . [1, 1, 2] = 3 from which t can be solved.
Substituting for t in will give the required point.	Solving for the value of t which satisfies; ([2, 1, 1] + t[0, 1, 2]) . [1, 1, 2] = 3 Carrying out the scalar (dot) product over the term in parentheses gives; [2, 1, 1] . [1, 1, 2] + t[0, 1, 2] . [1, 1, 2] = 3 2 + 1 + 2 + t + 4t = 3 t = - Substituting for t in the equations = [2, 1, 1] + t[0, 1, 2] = [2, 1, 1] + -[0, 1, 2] = [2 + 0, 1 + -, 1 + -] = [2, , ] The line therefore intersects the plane at the point (2, , )

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The Point of Intersection of a Line and a Plane

Problem:Find the point of intersection of the line having the position vector equation r1 = [2, 1, 1] + t[0, 1, 2] with the plane having the vector equation r2.[1, 1, 2] = 3

Problem:

Find the point of intersection of the line having the position vector equation r1 = [2, 1, 1] + t[0, 1, 2] with the plane having the vector equation r2.[1, 1, 2] = 3