1. Vectors (9) Lines in 3D Lines in 3D Angle between skew lines Angle between skew lines

2. x y z Skew lines a b In 3D lines can be that are not parallel and do not intersect are called skew lines Don’t meet

3. Skew Example 2 lines have the equations... and Show they are skew If the lines intersect, there must be values of s and t that give the position vector of the point of intersection. i : 2 + 4t = 4 +2s j : 3 - t = 7 - 2s k : 6 + 6t = 8 + s i+j : 5 + 3t = 11 3t = 6 t = 2 Substitutei : 2 + 4 x 2 = 4 +2s s = 3 Check the values in the 3rd equation k : 6 + 6 x 2 = 8 + 3 18 = 11 Not Satisfied! r = (2i + 3j + 6k) + t (4i - j + 6k) r = (4i + 7j + 8k) + s (2i - 2j + k) Direction vectors: (4i - j + 6k) and (2i - 2j + k) are not parallel Therefore lines are skew

4. Angles Between Skew Lines Skew lines do not meet! However you can work out angle between them by ‘transposing’ one to the other - keeping the direction the same. E.g. the angle between and You just need to look at the angle between the direction vectors: and

5. Skew Angle Example 2 lines have the equations … find the angle between them. and r = (2i + 3j + 6k) + t (4i - j + 6k) r = (4i + 7j + 8k) + s (2i - 2j + k) cos  = a.b |a||b| Direction Vectors are: a = 4i - j + 6k b = 2i - 2j + k a.ba.b |a||a| |b||b| =  (4 2 + -1 2 + 6 2 ) =  53 =  (2 2 + -2 2 + 1 2 ) =  9 = 3 = 4 x 2 + -1 x -2 + 6 x 1 = 16 cos  = 16 = 0. 3  53  = cos -1 (0.) = o

6. Angles Between Skew Lines - you find the angle! between and The direction vectors: and |a||a| =  (4 2 + -1 2 + 3 2 ) =  26 |b||b| =  (2 2 + -2 2 + 3 2 ) =  17 a.ba.b = 4 x 2 + -1 x -2 + 3 x 3 = 19  = cos -1 (0.904) = 25.3 o cos  = 19 = 0.904  26  17