Download Lines and Planes Linear algebra is the study of linearity in its most

1. Lines and Planes
1
Lines and Planes
Linear algebra is the study of linearity in its
most general algebraic forms. A collection of
mathematical objects exhibit linearity
• if, whenever X and Y are two such objects, they
can be added in a well-defined manner (that is,
X + Y makes sense as an object of the same type);
• if for all numbers a, the scalar multiple aX
makes sense as an object of the same type as X;
and
• for all such objects X and Y and numbers a,
scalar multiplication distributes over
addition:
a(X + Y) = aX + aY.
You have already experienced systems of
€
mathematical
objects that behave linearly, i.e.,
according to the properties listed above. The
prototypical examples of such systems are vectors
in the Euclidean plane R 2 or vectors in Euclidean
space R 3 (and, indeed, from any R n ). For this
reason, we call any system that satisfies the
defining properties
above (even if it is not of the
€
n
form
R
for some positive
€
€ integer n) a vector
space.
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1. Lines and Planes
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The centrality of the examples of R 2 and R 3 derives
from the fact that we have a geometric intuition
about their objects.
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€
Vectors in R 2 can be visualized as arrows of a
certain length and pointing in a certain direction
(the zero vector being an exception in that it has
zero length and no specified direction). This gives
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us a mental picture of what it means to add vectors
– via the parallelogram rule (see Figure 1.1.2, p. 2);
or, to multiply a vector by a scalar, which either
magnifies it – if the scalar has absolute value
larger than 1 – or shrinks it – if the scalar has
absolute value smaller than 1, while maintaining
its direction – if the scalar is positive – or reversing
direction – if the scalar is negative (see Figures
1.1.3, 1.1.4, p. 3).
The picture that we attach to vectors in R 2 – and
the situation in R 3 is much the same, with the
added complexity of a third dimension – makes it
clear why the subject is called linear
algebra: in R 2
€
(and in €
R 3 ), we can characterize a line L in terms of
vectors that satisfy the three properties we laid out
at the beginning of this discussion.
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€
1. Lines and Planes
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Where P and Q are any fixed vectors whose tips
lies on L, then Q – P is a vector that points in the
direction of L (and this is true regardless how P
and Q are chosen along L). Indeed, any two vectors
V and W that point in the direction of L have a sum
that also points in the direction of L; further, any
scalar multiple of such a vector points in the
direction of L, and all such vectors obey the
distributive property of scalar multiplication over
vector addition. Therefore, the set of scalar
multiples of Q – P forms a vector space.
It follows that all every vector X whose tip lies on L
has the form
X = P + t(Q – P)
or X = (1 – t)P + tQ, for some scalar t. So if we let t
run through all real values, these last equations
are equivalent forms for parametrizing the line
L. The variable scalar t here is called the
parameter of the equation.
1. Lines and Planes
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In R 2 , we can write out these vectors in coordinate
form: P = (p1, p2 ), Q = (q1,q2 ) are fixed vectors, and
X = (x, y) is a variable vector that depends on the
parameter t. Expanding via coordinates gives the
system of Cartesian parametric equations
€
€
€
x = p1 + t(q1 − p1 )
y = p2 + t(q2 − p2 )
for the line L. Since Q ≠ P (why not?), at least one
of the two coefficients of t above is not zero, so we
can solve€for t in that equation and substitute that
expression into the other equation; simplifying
produces a single equation of the form
ax + by + c = 0,
the Cartesian equation of the line L in the plane.
€
In R 3 the algebra is entirely similar, except that we
get a system of three Cartesian parametric
equations of the form
€
x = p1 + t(q1 − p1 )
y = p2 + t(q2 − p2 )
z = p3 + t(q3 − p3 )
€
1. Lines and Planes
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Eliminating the parameter as before produces a
pair of equations of the form
ax + by + cz + d = 0
a′x + b′y + c′z + d′ = 0
the Cartesian equations of the line L in space.
€ way we can describe equations of a
In a similar
plane Π in R 3 : where P, Q, R are fixed vectors
whose tips lie on Π and are not collinear (that is,
they don’t lie on the same line within Π), then
Q – P and R – P are vectors that point in different
€
€
directions within Π.
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€
It follows that any vector lying in Π can be
expressed as a sum of some scalar multiple of Q – P
€
and some scalar multiple of R – P.
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If X is any vector whose tip lies on Π, then X – P is
a vector lying in Π, so X – P = s(Q – P) + t(R – P)
for some pair of scalars s and t. Therefore,
€
€
X = P + s(Q – P) + t(R – P),
which parametrizes the vector equation of the
plane; here, both s and t are variable parameters.
(We could also write X = (1 – s – t)P + sQ + tR.)
1. Lines and Planes
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Writing this out in coordinate form produces a
system of Cartesian parametric equations for
the plane Π of the form
x = p1 + s(q1 − p1 ) + t(r1 − p1 )
€
y = p2 + s(q2 − p2 ) + t(r2 − p2 )
z = p3 + s(q3 − p3 ) + t(r3 − p3 )
Finally, treating this as a system of three equations
in the two unknowns s and t, we can use one of the
€
equations
to solve for one of the two variables,
substitute that expression into the other two
equations, and thereby obtain a system of two
equations in one unknown. Eliminating the
remaining parameter will produce a single
equation of the form
ax + by + cz + d = 0 ,
the Cartesian equation of the plane Π.
(Observe€again that the set of sums of scalar
multiples of Q – P with scalar multiples of R – P
€
forms a vector space!)