Vectors 6 Vector Equation of a Line Revise

Vectors (6) • Vector Equation of a Line

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

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

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

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

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

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

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

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

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

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

Convert this Vector Equation into Cartesian form r =[ Gradient = ] [] 7 +t 3 Increase in y 2 5 the direction vector Increase in x Gradient (m) = 5 / 2 = 2. 5 [ ] =[ ] x y 7 3 +t When t = 0 x 7 y = 3 [][] 2 5 x=7 y=3 Equations of form y= mx+c y= 2. 5 x + c 3 = 2. 5 x 7 + c c = -14. 5 y= 2. 5 x – 14. 5

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

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

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

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