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( Notice that different parameters are used. )

and. e.g. Determine whether the lines given below intersect. If they do, find the coordinates of the point of intersection. . ( Notice that different parameters are used. ). Solution:.

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( Notice that different parameters are used. )

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

  2. and If the lines intersect, there is a set of values for x, y and z that satisfy both equations. 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.

  3. l.h.s. r.h.s. So, the 3rd equation ( number (1) ) is not satisfied. The equation are said to be inconsistent. and x: y: 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): The lines do not intersect.

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

  5. 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. ( Never use all 3. ) • 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.

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

  7. For intersection, Solutions: 1(a) The lines are not parallel. I’ll 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.

  8. (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 2nd line or t in the 1st line: point of int. is ( 9, -9, 13 )

  9. Þ However, before we assume they don’t intersect, we should check whether they are collinear. (c) The line AB and the line CD where Solution: For AB, For CD, The lines are parallel.

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

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