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The Intersection of Lines

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( Notice that different parameters are used. ) and Solution: e.g. Determine whether the lines given below intersect. If they do, find the coordinates of the point of intersection. We notice first that the lines arent parallel. The direction vector of the 2 nd is not a multiple of the direction vector of the 1 st.

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The Intersection of Lines and With the left-hand sides of the equations equal, the right-hand sides will also be equal. This gives three equations, one for each component. If the lines intersect, there is a set of values for x, y and z that satisfy both equations.

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The Intersection of Lines x:x: y:y: z:z: There are 3 equations but we only need 2 of them ( the easiest ) to solve for the 2 unknowns. ( 2 ) gives t = 2 and ( 3 ) gives s = 1 Check in ( 1 ): So, the 3 rd equation ( number ( 1 ) ) is not satisfied. The equation are said to be inconsistent. l.h.s. r.h.s. The lines do not intersect. and

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The Intersection of Lines and x:x: y:y: z:z: ( 2 ) gives t = 2 and ( 3 ) now gives s = 0 Check in ( 1 ): All 3 equations are now satisfied so the lines intersect. l.h.s. r.h.s. If the equation of the 3rd line is changed: We now get The point of intersection is found by substituting for s or t.

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The Intersection of Lines SUMMARY Since a point of intersection ( x, y, z ) would lie on both lines: Equate the right-hand sides of the equations of the lines. Write down the 3 component equations. Solve any 2 equations. Try to pick the easiest. Check whether the values of s and t satisfy the unused equation. If not, the lines are skew. If all equations are satisfied, substitute into either of the lines to find the position vector or coordinates of the point of intersection. ( Never use all 3. )

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The Intersection of Lines Exercise 1. Determine whether the following pairs of lines intersect. If they do, find the coordinates of the point of intersection. (a) (b) (c) The line AB and the line CD where and

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The Intersection of Lines Solutions: 1(a) The lines are not parallel. For intersection, Ill use ( 1 ) and ( 2 ): Subs. in ( 2 ): Check in ( 3 ): l.h.s. 1, r.h.s. 2 The equations are inconsistent. The lines are skew.

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The Intersection of Lines (b) Solution: The lines are not parallel. For intersection, Subs. in (2) : Check in (3) : l.h.s. = 13, r.h.s. = 13 Lines meet. Subs. for s in the 2 nd line or t in the 1 st line: point of int. is ( 9, 9, 13 )

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The Intersection of Lines (c) The line AB and the line CD where Solution: For AB,For CD, The lines are parallel. However, before we assume they dont intersect, we should check whether they are collinear.

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The Intersection of Lines We could have either or no points of intersection an infinite number of points of intersection x x A B x x D C x x x x A B C D We can check by seeing whether C, or D, lies on AB. equation of AB is does not lie on this line. The lines dont intersect.

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