Work Sheet 5—Equations of Lines
_______________________
NAME
INSTRUCTIONS: You may refer to Section 12.5 Equations of Lines and Planes to answer
the following questions. You may discuss it with a classmate.
DEFINITION:
A line L in 3-space is determined by a point P0(x0, y0, z0) on the line L
and the direction of the line provided by a vector v
r = r0 + tv
Where t is a parameter which runs from -∞ to ∞ to trace out the line. This is the vector
equation for the line L.
If v, the direction of the line L, v = {a, b, c}. The position vector to a known point on the line
L is r0 = {x0, y0, z0}. The general position vector is r = {x, y, z} whose tip traces out the
line. Then vector addition gives
{x, y, z} = {x0 + t a, y0 + t b, z0 + t c}
This corresponds to the three corresponding scalar equations:
x = x0 + t a
y = y0 + t b
z = z0 + t c
These equations are called the parametric equations of the line.
Question 1
(a)
Determine the vector equation of the line passing through (1, 2, -3) and parallel to v =
4i + 5j – 7k.
(b)
Write the parametric equations for this line.
Question 2
Consider a line L through the points P1(2, 4, -1) and P2(5, 0, 7).
(a)
Find a vector, v, parallel to this line.
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(b)
Write parametric equations for this line.
(c)
To find the point of intersection of this line with the xy-plane, use the z parametric
equation to determine the t value at the intersection.
(d)
What are the (x, y, z) coordinates of the point of intersection?
Symmetric Equations of a Line
Starting with the parametric equations
x = x0 + t a
y = y0 + t b
z = z0 + t c
Solve each equation for t and equate the results to obtain symmetric equations
x − x 0 y − y 0 z − z0
=
=
a
b
c
The denominators form the components of the direction vector, v = {a, b, c}
Question 3
Given the line, L:
x −1 y z −1
= =
.
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1
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Find the point/s of intersection, if any, with the coordinate planes
(a)
With the xy-plane , (x, y, z) = _____________
(b)
With the xz-plane , (x, y, z) = _____________
(c)
With the yz-plane , (x, y, z) = _____________
Question 4
Determine whether lines L1 and L2 are parallel. Justify your conclusion.
L1 :
x = 3 – 2t, y = 4 + t, z = 6 – t
L2 :
x = 5 – 4t, y = -2 + 2t, z = 7 – 2t
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Question 5
Show that the lines L1 and L2 are the same line. Justify your conclusion.
L1 :
x = 1 + 3t, y = -2 + t, z = 2 t
L2 :
x = 4 – 6t, y = -1 - 2t, z = 2 – 4t
Intersection of two Lines
Given two lines L1 and L2:
L1 :
x = g(t), y = h(t), z = k(t)
L2 :
x = m(s), y = n(s), z = p(s)
Procedure:
(i)
Equate g(t) = m(s), h(t) = n(s) and k(t) = p(s)
(ii)
Solve for s and t.
(iii)
If unique values of t and s, insert these values into the equations for x, y, z to
determine the coordinates of the point of intersection.
(iv)
If unique values of t and s can not be found, the lines do not intersect.
Question 6
Let L1 and L2 be the lines whose parametric equations are:
L1 :
x = 1 + 2t, y = 2 - t, z = 4 – 2t
L2 :
x = 9 + s, y = 5+3s, z = -4 – s.
(a)
Find the point of intersection.
(b)
Find the acute angle between the lines L1 and L2 at their intersection.
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(c)
Find parametric equations for the line that is perpendicular to L1 and L2 and passes
through their point of intersection.
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