Recall
Equation for a line
• Scalar equation (only in 2-D) : Ax + By + C = 0
• Vector equation:
~r = r~0 + tm,
~
t∈R
where r~0 is the position vector of any point on the line and m
~ is a
direction vector for the line.
• Parametric equation: x = x0 + tm1 , y = y0 + tm2 , t ∈ R
Two lines L1 and L2 are parallel if they have the same slope. That means,
in vector equation form, their direction vectors are parallel.
Two lines L1 and L2 are coincident if they are the same line.
Example:
Which two lines are parallel?
(a) L1 ~r = (2 − 3t)ı̂ + (1 − 7t)̂ and L2 ~r = (1 − 6t)ı̂ + (4 + t)̂
(b) L1 ~r = [0, 3, −1] + t[1, 2, −2] and L2 ~r = [0, 6, 3] + t[3, 9, 1]
(c) L1 ~r = (2 + 3t)ı̂ + (1 − 2t)̂ and L2 ~r = (6 − 6t)ı̂ + (4 + 4t)̂
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Given a line L, a normal vector is any vector ~n that is perpendicular to the
direct vector m
~ of the line (i.e., m
~ · ~n = 0).
This means a normal vector is perpendicular to the line L too. (Why?)
Example:
The scalar equation of a line L is y = 2x + 6. Show that ~n = [2, −1] is normal
to L.
Given the scalar equation Ax + By + C = 0 for a line, the vector ~n = [A, B]
is normal to it.
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Equations of Lines (3-D)
• There is NO scalar equation for a line in 3-dimensional space, NEITHER do we have a slope-intercept form.
• A scalar equation in three variables Ax + By + Cz + D = 0 describes
a plane in three-space, not a line.
• vector equation: The line passing through P0 = (x0 , y0 , z0 ) with direction vector m
~ = [m1 , m2 , m3 ] is
~r = r~0 + tm,
~ t∈R
or
[x, y, z] = [x0 , y0 , z0 ] + t[m1 , m2 , m3 ]
• parametric equations:
x = x0 + tm1 , y = y0 + tm2 , z = z0 + tm3 for t ∈ R
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Example:
(a) Find the vector equation and parametric equation of the line L1 passing
through P1 = (1, 0, 5) and P2 = (4, 2, 1).
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Intersection of Lines For two lines L1 and L2 in two-space, there are three
possibilities.
• (i) The lines intersect at a single point.
• (ii) The lines are coincident (and hence intersect at infinitely many
points).
• (iii) The lines are parallel and distinct (and hence do not intersect).
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If we have two lines L1 and L2 in three-space, there are four possibilities.
• (i) The lines intersect at a single point (unique solution).
• (ii) The lines are coincident (there are infinitely many solutions).
• (iii) The lines are parallel and distinct (no solution).
• (iv) The lines are skew – they are distinct, not parallel and do not
intersect (no solution).
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Example:
(a) Find the point of intersection of L1 [x, y, z] = [2, 3, 4] + t[1, 1, 3] and
L2 [x, y, z] = [1, 0, −1] + s[1, 1, 3] (if any).
(b) Find the point of intersection of L1 ~r = [2, 1, 5] + t[3, 1, −2] and L2 ~r =
[0, 3, 1] + s[5, −1, 2].
(c) Find the point of intersection of L1 [x, y, z] = [2, 1, 5] + t[3, 1, −2] and
L2 [x, y, z] = [1, 3, 2] + s[5, −1, 2].
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The distance between two skew lines
The shortest distance between two skew lines L1 and L2 is the length of the
~
common perpendicular |AB|.
Suppose we have points P1 and P2 on the lines. Then if ~n is any vector in
~ the shortest distance between two lines L1 and L2 is
the direction of AB,
~
~ = |proj~n P1~P2 | = |P1 P2 · ~n|
|AB|
|~n|
Example:
Find the distance between L1 [x, y, z] = [2, 1, 5] + t[3, 1, −2] and L2 [x, y, z] =
[1, 3, 2] + s[5, −1, 2].
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Equation of a plane
Direction vectors of a plane are two non-parallel vector on the plane (or
parallel to the plane).
A vector that is perpendicular to a plane is called a normal vector of the
plane.
• The vector equation of a plane is
[x, y, z] = [x0 , y0 , z0 ] + t~a + s~b = [x0 , y0 , z0 ] + t[a1 , a2 , a3 ] + s[b1 , b2 , b3 ]
or
~r = r~0 + t~a + s~b
where r~0 is a position vector of a point on the plane and ~a and ~b are
direction vectors for the plane.
• The parametric equation of a plane is
x = x0 + ta1 + sb1 ,
y = y0 + ta2 + sb2 ,
z = z0 + ta3 + sb3
where t, s ∈ R.
• The scalar equation of a vector is
Ax + By + Cz + D = 0,
where ~n = [A, B, C] is a normal vector of the plane.
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Example:
Write the vector equation and parametric equation of the plane passing
through points A(1, 2, 3) and B(−1, 2, 1) and C(3, 1, 4).
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Example:
Let’s find the equation of the plane that contains the line [x, y, z] = [2, 1, 4] +
t[2, 3, 4] and is parallel to the line [x, y, z] = [7, 4, 2] + s[−1, 0, 6].
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