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Vectors (9)

Vectors (9). Lines in 3D Angle between skew lines. 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. Skew Example. Check the values in the 3rd equation. 2 lines have the equations.

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Vectors (9)

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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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