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Vector Equations of Straight Lines in 2D and 3D Geometry

Learn about vector equations of straight lines, position vectors, parallel vectors, and conversions to Cartesian form with clear examples.

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Vector Equations of Straight Lines in 2D and 3D Geometry

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  1. Vectors (6) Vector Equation of a Line

  2. Revise: Position Vectors a = xi + yj + zk z A In 2D and 3D, all points have position vectors a y e.g. The position vector of point A o x

  3. -20 i + 30 j -10 i + 15 j 2 a a -2 i + 3 j 1/5a Revise: Parallel Vectors Vectors with a scaler applied are parallel i.e. with a different magnitude but same direction

  4. Vector Equation of a line (2D) A a Any parallel vector (to line) (any point it passes through) y o x A line can be identified by a linear combination of a position vector and a free vector

  5. Vector Equation of a line (2D) y Any parallel vector to line A a (any point it passes through) o x A line can be identified by a linear combination of a position vector and a free vector

  6. Vector Equation of a line (2D) t is a scaler - it can be any number, since we only need a parallel vector a = xi+ yj b = pi+ qj A line can be identified by a linear combination of a position vector and a free vector y A parallel vector to line x o E.g. a + tb = (xi+ yj) + t(pi+ qj)

  7. [ ] [ ] [ ] 2 2 2 2 2 2 [ ] [ ] [ ] 1 3 1 3 1 3 [ ] x y Scaler (any number) Vector Equation of a y = mx + c (1) 1. Position vector to any point on line y = x + 2 2. A free vector parallel to the line 3.linear combination of a position vector and a free vector = + t Equation

  8. [ ] [ ] [ ] -3 -3 -3 -3 -3 -3 [ ] [ ] [ ] 4 6 4 6 4 6 = + t [ ] x y Scaler (any number) Vector Equation of a y = mx + c (2) 1. Position vector to any point on line y = x + 2 2. A free vector parallel to the line 3.linear combination of a position vector and a free vector Equation

  9. [ ] [ ] [ ] 4 2 4 2 4 2 [ ] [ ] [ ] 2 4 2 4 2 4 [ ] x y Scaler (any number) Vector Equation of a y = mx + c (3) 1. Position vector to any point on line y = 1/2 x + 3 2. A free vector parallel to the line 3.linear combination of a position vector and a free vector = + t Equation

  10. [ ] [ ] [ ] [ ] [ ] 1 2 1 2 1 2 When t=0 1 3 1 3 = x=1, y=2 When t=1 [ ] 2 5 = [ ] [ ] [ ] x y x y x y x=2, y=5 Sketch this line and find its equation y = 3x - 1 = + t

  11. [ ] [ ] [ ] [ ] 1 2 1 2 1 3 1 3 = + t r = + t [ ] x y Equations of straight lines y = 3x - 1 ….. is a Cartesian Equation of a straight line ….. is a Vector Equation of a straight line Often written ……. r is the position vector of any point R on the line Any point Direction

  12. [ ] [ ] [ ] [ ] [ ] 2 5 2 5 r 7 3 7 3 7 3 = + t the direction vector = + t Increase in y Gradient = Increase in x [ ] [ ] = x y x y Convert this Vector Equation into Cartesian form Gradient (m) = 5 / 2 = 2.5 Equations of form y= mx+c y= 2.5x + c When t = 0 3 = 2.5 x 7 + cc = -14.5 x = 7 y = 3 y= 2.5x – 14.5

  13. [ ] [ ] [ ] [ ] 2 5 2 5 r 7 3 7 3 = + t = + t [ ] x y Convert this Vector Equation into Cartesian form (2) x = 7 + 2t y = 3 + 5t Convert to Parametric equations 5x = 35 + 10t 2y = 6 + 10t Eliminate ‘t’ 5x – 2y = 29 subtract

  14. [ ] [ ] a b 1 m r = + t [ ] Any point 0 3 the direction vector = 4 1 Increase in y Gradient = [ ] Increase in x [ ] [ ] 1 4 0 3 1 4 r = + t Convert this Cartesian equation into a Vector equation Want something like this ………. y = 4x + 3 When x=0, y = 4 x 0 + 3 = 3 = Any point Gradient (m) = 4 represents the direction

  15. Can replace with a parallel vector [ ] [ ] [ ] 0 3 x y 1/4 1 = + t [ ] [ ] 0 3 1 4 r = + t Convert this Cartesian equation into a Vector equation Easier Method y = 4x + 3 y - 3 = 4x = t Write: x = 1/4 t t = 4x t = y - 3 y = 3 + t

  16. [ ] [ ] a b 1 m r = + t Any point the direction vector Summary A line can be identified by a linear combination of a position vector and a free [direction] vector Equations of form y-b=m(x-a) Line goes through (a,b) with gradient m

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