The Point of Intersection of a Line and a Plane |
Problem: |
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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. |
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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. |
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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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