Vectors (6)Vectors (6)•Vector Equation of a LineVector Equation of a Line

x

y

z

a

Revise: Position Vectors

o

A In 2D and 3D, all points have position vectors

e.g. The position vector of point A

z

y

x

a

a = xi + yj + zk

Revise: Parallel VectorsRevise: Parallel Vectors

a

-10 i + 15 j 2 a

-20 i + 30 j

1/5 a

-2 i + 3 jVectors with a scaler applied are parallel

i.e. with a different magnitude but same direction

x

y

Vector Equation of a line (2D)

o

A line can be identified by a linear combinationof a position vector and a free vector

Any parallel vector (to line)a(any point it passes through)

A

x

y

a

Vector Equation of a line (2D)

o

(any point it passes through)

A line can be identified by a linear combinationof a position vector and a free vector

A

Any parallel vector to line

x

y

Vector Equation of a line (2D)

o

A line can be identified by a linear combinationof a position vector and a free vector

A

parallel vector to linea = xi + yj

b = pi + qj

E.g. a + tb

= (xi + yj) + t(pi + qj)

t is a scaler- it can be any number, since we only need a parallel vector

Vector Equation of a y = mx + c (1)

y = x + 21. Position vector to any point on line

13[ ]

13[ ]

2. A free vector parallel to the line

22[ ]

22[ ]

3. linear combination of a position vector and a free vector

xy[ ]= + t1

3[ ] 22[ ]

Equation Scaler (any number)

Vector Equation of a y = mx + c (2)

y = x + 21. Position vector to any point on line

2. A free vector parallel to the line

3. linear combination of a position vector and a free vector

Equation Scaler (any number)

46[ ]

46[ ] -3

-3[ ]-3-3[ ]

xy[ ]= + t4

6[ ] -3-3[ ]

Vector Equation of a y = mx + c (3)

y = 1/2 x + 3

1. Position vector to any point on line

2. A free vector parallel to the line

3. linear combination of a position vector and a free vector

Equation Scaler (any number)

xy[ ]= + t

24[ ]

24[ ]

42[ ]

42[ ]

42[ ]2

4[ ]

Sketch this line and find its equation

y = 3x - 1xy[ ]= + t 1

3[ ]12[ ]

12[ ]

13[ ]1

2[ ]

=

When t=1

xy[ ]

When t=0

xy[ ] =

25[ ]

x=1, y=2

x=2, y=5

y = 3x - 1….. is a Cartesian Equation

of a straight line

xy[ ]= + t 1

3[ ]12[ ]

….. is a Vector Equation of a straight line

Often written …….

= + t13[ ]1

2[ ]r r is the position vector of any point R on the line

Equations of straight lines

Any point

Direction

Convert this Vector Equation into Cartesian form

= + t25[ ]7

3[ ]r

xy[ ]= + t

73[ ] 2

5[ ]

Increase in yIncrease in x

Gradient =

the direction vector

Gradient (m) = 5 / 2 = 2.5

When t = 0

xy[ ] 7

3[ ]= x = 7y = 3

Equations of form y= Equations of form y= mx+c mx+c

y= 2.5x + c y= 2.5x + c

3 = 2.5 3 = 2.5 xx 7 + c 7 + cc = -14.5 c = -14.5

y= 2.5x – 14.5 y= 2.5x – 14.5

Convert this Vector Equation into Cartesian form (2)

= + t25[ ]7

3[ ]r

xy[ ]= + t

73[ ] 2

5[ ]x = 7 + 2ty = 3 + 5t

Convert toParametric equations

Eliminate ‘t’5x = 35 + 10t2y = 6 + 10t

subtract 5x – 2y = 29

Convert this Cartesian equation into a Vector equation

14[ ]

Increase in yIncrease in x

Gradient =

Gradient (m) = 4

y = 4x + 3 y = 4x + 3 = + t1m[ ]a

b[ ]r

the direction vector

Any point

Want something like this ……….

When x=0, y = 4 x 0 + 3 = 3

[ ]03

= Any point

= 4 1

representsthe direction

= + t14[ ]0

3[ ]r

Convert this Cartesian equation into a Vector equation

y = 4x + 3 y = 4x + 3

Easier Method

Write: y - 3 = 4x = t y - 3 = 4x = t

t = 4x t = y - 3

x = 1/4 ty = 3 + t

xy[ ]= + t

03[ ] 1/4

1[ ]

= + t14[ ]0

3[ ]r

Can replace with a parallel vector

Summary

= + t1m[ ]a

b[ ]r

the direction vector

Any point

A line can be identified by a linear combinationof a position vector and a free [direction] vector

x

y

o

Any parallel vector (to line)

a

(any point itpasses through)A

Equations of form y-Equations of form y-b=m(x-a) b=m(x-a)

Line goes through (a,b) with gradient m