Chapter 8: Equations of Lines and Planes
We have spent the last two chapters looking at vectors. In many ways, they seem similar to lines. In fact, lines are not vectors, but vectors are used to describe lines.
Similarities and Differences of LInes and Vectors
Lines
Vectors
Lines are bi­directional. A line defines Vectors are unidirectional. A vector a direction, but there is nothing to defines a direction with a clear distinguish forward from backward.
distinction between forward and backward.
A line is infinite in both directions. A A vector has a finite magnitude.
line segment has a finite length.
Lines and line segments have a A vector has no fixed location. The definite location. The opposite sides opposite sides of a parallelogram are of a parallelogram are two different described by the same vector.
line segments.
Two lines are the same when they Two vectors are the same when they have the same direction and same have the same direction and the same location. These lines are said to be magnitude. These vectors are said to coincident.
be equal.
Consider y = mx + b
slope = direction of the line
y ­ intercept = starting point
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Section 8.1: Vector and Parametric Equations in R2
To find the vector and parametric equations of a line in R2, we must have a direction vector.
A direction vector is a nonzero vector m = (a, b) parallel (collinear) to the given line. It is represented by a vector with its tail at the origin and its head at the point (a, b)...sound familiar?
The x and y components of this direction vector are the direction numbers.
Example 1
Representing Lines Using Vectors
a) A line passing through P(4,3) has m = (­7, 1) as its direction vector. Sketch this line.
b) A line passes through the points A(1/2, ­3) and B(3/4, 1/2). Determine a direction vector for this line and write it using integer components. 2
Parametric and Vector Equations of a LIne in R2
Vector Equation: r = r0 + t m, t R
Parametric Equation: x = x0 + ta, y = y0 + tb, t R
where r0 is the vector from (0,0) to the point (x0, y0) and m is the direction vector with components (a,b).
In either parametric or vector form, t, is called a parameter. This means that t can be replaced by any real number to obtain coordinates of points on the line.
Example 2
a) Determine the vector and parametric equations of a line passing through point A(1,4) with direction vector m = (­3, 3).
b) Sketch the line and determine the coordinates of four points on the line.
c) Is either point Q(­21, 23) or point R(­29, 34) on this line?
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Example 3
a) Determine vector and parametric equations for the line containing points E(­1,5) and F(6,11).
b) What are the coordinates of the point where this line crosses the x­axis?
c) Can the equation r = (­15,­7) + t(14/3,4). t R, also represent the line containing points E and F?
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Example 4
Determine a vector equation for the line that is perpendicular to r = (4,1) + s(­3, 2), s R, and passes through point P(6,5).
Homework
p. 433­434 # 1­6, 8­12
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Section 8.3 Vector and Parametric Equations in R3
­ basically the same as R2 ­ parametric ­ one more equation for z component
Symmetric Equations of Lines in R3
­ use the parametric equations and solve for the parameter in each component x = x0 + ta
y = y0 + tb z = z0 + tc
t = x ­ x0
t = y ­ y0
t = z ­ z0
a
b
c
Example 5
Write the symmetric equations of a line passing through the points A(­1, 5, 7) and B(3, ­4, 8).
Homework p. 449­450 #1, 3, 4, 5bdf, 6, 7, 8, 10 ac, 12
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