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

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Presentation transcript:

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

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

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 : x 2 = 4 +2s s = 3 Check the values in the 3rd equation k : x 2 = = 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

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

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| =  ( ) =  53 =  ( ) =  9 = 3 = 4 x x x 1 = 16 cos  = 16 = 0. 3  53  = cos -1 (0.) = o

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