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Angles Between Skew Lines in 3D Space: Helpful Guide

Understand skew lines in 3D space, calculate angles between them, and solve for intersection points. Step-by-step examples provided to grasp concepts easily.

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Angles Between Skew Lines in 3D Space: Helpful Guide

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

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

  3. Skew Example Check the values in the 3rd equation 2 lines have the equations ... r= (2i + 3j + 6k) +t(4i - j + 6k) Show they are skew r= (4i + 7j + 8k) +s(2i - 2j + k) and If the lines intersect, there must be values of s and tthat give the position vector of the point of intersection. i: 2 + 4t = 4 +2s j: 3 -t = 7 - 2s k: 6 + 6x2 = 8 + 3 18 = 11 k: 6 + 6t = 8 + s Not Satisfied! i+j: 5 + 3t = 11 3t = 6 t = 2 Direction vectors: (4i - j + 6k) and (2i - 2j + k) are not parallel Therefore lines are skew Substitutei: 2 + 4x2 = 4 +2s s = 3

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

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

  6. Angles Between Skew Lines - you find the angle! The direction vectors: between and and a.b = 4x2 + -1x-2 + 3x3 = 19 |a| = (42 + -12 + 32) = 26 |b| = (22 + -22 + 32) = 17 cos = 19 = 0.904 2617  = cos-1(0.904) = 25.3o

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