Download Review of Lines in the Plane • increments A

Review of Lines in the Plane
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increments
A particle moves from the point (x1, y1) to the point (x2, y2).
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The resulting increment (change) in x, denoted Δx, is given by:
Δx = x2 – x1 .
Similarly, the resulting increment in y, denoted Δy, is given by:
Δy = y2 – y1 .
slope of a line
Let L be a nonvertical line joining the two points (x1, y1) and (x2, y2).
The slope of L is: m =
y −y
rise
Δy
=
= 2 1.
Δx
run
x 2 − x1
The slope of a line in the usual variables x and y is the change in y per unit change
in x - the change in y with respect to x.
So Δy = mΔx.
(e.g. If the slope m = 4, then a change in x of 3 units forces a change in y of 4×3 =12
units.)
The slope is positive if x and y change in the same direction and negative if they
change in opposite directions.
A line is horizontal if it exhibits zero change in y per unit x, that is it has the three
equivalent properties:
1. Δy = 0 for Δx ≠ 0;
2. the slope m = 0;
3. y is constant over all values of x.
A line is vertical if no change in x is possible, that is it has the three equivalent
properties:
1. Δx = 0 for Δy ≠ 0;
2. the slope m is undefined;
3. x constant over all values of y.
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parallel and perpendicular lines
Two lines L1 and L2 are said to be parallel if they form equal angles with the x – axis.
That is, they have equal slopes (m1 = m2).
Two lines L1 and L2 are said to be perpendicular if their slopes are negative
−1
reciprocals of one another. That is, m1 =
.
m
2
L1
L2
a
b
θ2
θ1
Note: The slope of L1 is m1 = tan θ1 = a/b. But m2 = tan θ2 = −b/a. So m1 = −1/m2 .
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equation of a line
Consider a line L passing through a point (x1, y1). Let (x, y) be any other point on L.
y - y1
or y – y1 = m(x – x1).
x - x1
The equation of a line has three common forms:
Then the slope m of L is: m =
1. y – y1 = m(x – x1);
2. y = mx + b
3. Ax + By = C, m =
4. x/a + y/b = 1
−A
,
B
the point – slope form
the slope intercept form
the general form for A, B ≠ 0
intercept form if a and b are
the non zero x and y
intercepts
Graphing a linear equation:
1. y = a ⇒ a horizontal line through (0, a).
2. x = a ⇒ a vertical line through (a, 0).
3. m ≠ 0 plot using x and y intercepts. That is, x = 0 ⇒ (0, b) is on the line where b
is the y- intercept (the y coordinate of the point where the line crosses the y axis)
and y = 0 ⇒ (a, 0) is on the line where a is the x-intercept (the x coordinate of the
point where the line crosses the y axis).
4. If the line passes through the origin (0, 0) then use the slope to plot the line. One
can always use a known point and slope to plot a line.
Example 1:
Find the equation of the line through the points (1, 4) and (−2, 3).
solution: 1. Find the slope: m =
3 − 4 −1 1
=
= .
−2 − 1 −3 3
2. Substitute the slope and one of the two points into the point slope
equation of the line.
3. y − 4 = (1/3)(x − 1)
3y − 12 = x − 1
−x + 3y − 11 = 0 or x − 3y + 11 = 0.