Let L be the line that passes through the point x 0 , y 0 , z 0 and is parallel to the vector v = a , b , c , where a , b , and c are nonzero. Show that a point x , y , z lies on the line L if and only if x − x 0 a = y − y 0 b = z − z 0 c These equations, which are called the symmetric equations of L , provide a nonparametric representation of L .
Let L be the line that passes through the point x 0 , y 0 , z 0 and is parallel to the vector v = a , b , c , where a , b , and c are nonzero. Show that a point x , y , z lies on the line L if and only if x − x 0 a = y − y 0 b = z − z 0 c These equations, which are called the symmetric equations of L , provide a nonparametric representation of L .
Let L be the line that passes through the point
x
0
,
y
0
,
z
0
and is parallel to the vector
v
=
a
,
b
,
c
,
where a, b, and c are nonzero. Show that a point
x
,
y
,
z
lies on the line L if and only if
x
−
x
0
a
=
y
−
y
0
b
=
z
−
z
0
c
These equations, which are called the symmetric equations of L, provide a nonparametric representation of L.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Determine a vector equation for each line.
a) perpendicular to line 4x - 3y = 17 and through point P(- 2, 4)
b) parallel to the z-axis and through point P(1, 5, 10)
c) parallel to [x, y, z] = [3, 3, 0] + t[3, 5, 9] with x-intercept of - 10.
-
d) with the same x-intercept as [x, y, z] = [3, 0, 0] + t[4, 4, 1] and the same z-intercept as [x, y,
z]= [6, 2, 3] + [3, -1, -2].
I
16. Determine the vector equation of
each line.
a) parallel to the y-axis and through
Po(-4, 11)
b) perpendicular to 5x - 2y = 3 and
through Po(7, 1)
c) parallel to x + 6y = 1 and through
Po(2,-5)
d) parallel to the z-axis and through
Po(-4, 3, 7)
e) parallel to
[x, y, z]= [1, 3, -2] + [5, 2, 1]
f) the x-intercept of the line is 3 and the
y-intercept is -2
Let / denote the line in R² with equation y = 3x - 2. Let Q = (2, -1).
(a) Write the line / in vector form (i.e. of the form p = po + td where Po is a point on the line and d is a direction
vector).
(b) Find the distance from Q to 1.
(c) Find the point on I closest to Q.
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