Scalar Equation Of A Line Worksheet With Answers - 2011 Page 5
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112. Scalar Equation of a Line.notebook
May 19, 2011
So, 3 x + 5 y 7 = 0 is a scalar equation of the line through the point P
(4, 1)
0
with normal n = (3, 5).
Notice the location of the components of the normal vector 3 and 5 in
the scalar equation as coefficients of the x and y terms. This is not a
coincidence but a general result.
A scalar equation of the line through point P
( x
, y
) with normal
0
0
0
n = ( n
, n
is given by
)
1
2
n
x + n
y + C =0
1
2
For point P( x , y ) on the line, vector P
P is in the direction
Proof:
0
of the line.
y
P
P • n = 0
0
( x x
, y y
) • ( n
, n
) = 0
0
0
1
2
P
( x
, y
)
0
0
0
n
( x x
) + n
( y y
) = 0
1
0
2
0
( n
, n
)
P( x , y )
1
2
n
x n
x
+ n
y n
y
= 0
1
1
0
2
2
0
n
x + n
y n
x
n
y
= 0
1
2
1
0
2
0
x
The expression n
x
n
y
is a constant.
1
0
2
0
∴ n
x + n
y + C =0
1
2
5
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