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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