Vectors (9)Lines in 3DAngle between skew lines

Skew linesabIn 3D lines can be that are not parallel and do not intersect are called skew linesDont meet

Skew Example2 lines have the equations ...andShow they are skewIf 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 +2sj : 3 - t = 7 - 2sk : 6 + 6t = 8 + si+j : 5 + 3t = 11 3t = 6 t = 2Substitutei : 2 + 4x2 = 4 +2s s = 3k : 6 + 6x2 = 8 + 3 18 = 11Not 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 parallelTherefore lines are skew

Angles Between Skew LinesSkew lines do not meet!

However you can work out angle between them bytransposing one to the other - keeping the direction the same.

Skew Angle Example2 lines have the equations find the angle between them. andr = (2i + 3j + 6k) + t(4i - j + 6k)r = (4i + 7j + 8k) + s(2i - 2j + k)Direction Vectors are:a = 4i - j + 6kb = 2i - 2j + ka.b|a||b| = (42 + -12 + 62) = 53 = (22 + -22 + 12) = 9 = 3= 4x2 + -1x-2 + 6x1 = 16cos = 16 = 0. = cos-1(0.) = o

Angles Between Skew Lines - you find the angle! = cos-1(0.904) = 25.3o