Flashcards
Parallel, Intersecting, Skew and same lines
Cards in this deck (9)
- 1In 2D and 3D, parallel lines have how many solutions?None (as they don't intersect)
- 2In 2D and 3D, non-parallel lines have how many solutions?1 (as they intersect once)
- 3In 2D and 3D, 2 identical (same) lines have how many solutions?Infinitely many
- 4In 3D, how many solutions does a skew line have?0 (as no intersection)
- 5When we use a simultaneous equation to determine points of intersection, we must remember to use....Different parameter letters
- 6Give an example of a line that is parallel to r = [1][0][2] + λ [2][-3][0]r = [1][0][3] + λ [2][-3][0]
(gradient same, starting vector slightly different) - 7Determine whether r = [3][1] + λ [-1][2] and r = [4][1] + t [2][-3] intersect and if they do find the point of intersectionGradients are not parallel and therefore intersect at a point
3-λ = 4+2t , -λ-2t = 1
1+2λ = 1-3t , 2λ+3t = 0
Simultaneously solve -λ-2t=1 and 2λ+3t=0 to get λ=3 and t=-2
Sub in: r = [3][1] + 3 [-1][2] = [0][7]
Therefore intersect at (0,7) - 8Determine whether r = [-1][6] + λ [-3][2] and r = [2][2] + t[6][-4] are parallel, intersect or are the same line[-3][2] = -2[6][-4] so parallel
Check to see if same line or not:
[2][2] = [-1][6] + λ [-3][2] ;
2 = -1-3λ , λ=-1
2 = 6+2λ , λ=-2
Therefore not same line as λ not consistent, and just parallel - 9Determine whether r = [-1][3][2]+λ[1][-2][5] and r = [5][-2][1] + t [2][3][-1] are parallel, skew or intersectNot parallel (as direction vectors different)
[-1][3][2]+λ[1][-2][5] = [5][-2][1] + t [2][3][-1]
-1+λ=5+2m , λ-2t=6
3-2λ=-2+3t , -2λ-3t=-5
Solve simultaneously λ-2t=6 and -2λ-3t=-5 to get λ=4 and t=-1
Sub into final equation ; 2+5λ=1-t , 2+5(4)=1-(-1) , 22 ≠ 2 so therefore the lines don't intersect, so are skew