The Point of intersection of two Lines |
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 two lines. and are the position vectors of any point on the respective lines. |
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The position vectors are shown drawn at the point of intersection of the two lines where = . | Since, at the point of intersection, the two position vectors are identical it follows that; [0, 0, 1] + t[1, -1, 1] = [4, 1, 2] + s[-6, -4, 0] From which t and s can be solved. |
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Substituting for t in or, s, in will give the required point. | Solving for the value of s and t which satisfies; [0, 0, 1] + t[1, -1, 1] = [4, 1, 2] + s[-6, -4, 0] Rearranging; t[1, -1, 1] - s[-6, -4, 0 ] = [4, 1, 2] - [0, 0, 1] [ t - -6s, -t - -4s, t ] = [4, 1, 1] Equating components; t + 6s = 4 1. -t + 4s = 1 2. t = 1 3. Equations 2. and 3. inply; t = 1 s = Substituting in the equations for = [0, 0, 1] + t [1, -1, 1] or, = [4, 1, 2]+ [-6, -4, 0] gives; = [0, 0, 1] + 1 [1, -1, 1] = [4, 1, 2]+ [-6, -4, 0] = [0 + 1, 0 + -1, 1 + 1] = [4 + -3, 1 + -2, 2 + 0] = [ 1,-1, 2] = [1, -1, 2] The two lines therefore intersect at the point (1, -1, 2) |
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