Course : MATH6030001 - Linear Algebra

Effective Period : February 2024

Vectors

Session 07-10
Acknowledgement
These slides have been
adapted from:

Anton, Howard, and Chris

Rorres. Elementary linear
algebra: applications
version (12th Edition). John
Wiley & Sons, 2019.

Chapter 3, Section 4.3 - 4.6

 Learning Objectives
LO 3 : Explain vector operations and their geometric interpretations

• Students will be able to

compute various vector
operations.
• Students will be able to interpret
vector operations geometrically.
 Contents
Vectors in 2-Space and 3-Space

Norm, Dot Product, and Distance in Rn

Orthogonal Projection in Rn
Linear combination, linear
independence, and basis in Rn

Cross Product in R3

Lines and planes equation in R3

Vectors in 2-Space and 3-Space
 Vectors in 2-Space and 3-Space
Engineers and physicists represent vectors in two dimensions (also
called 2-space) or in three dimensions (also called 3-space) by arrows.
The direction of the arrowhead specifies the direction of the vector and
the length of the arrow specifies the magnitude. Mathematicians call
these geometric vectors. The tail of the arrow is called the initial point
of the vector and the tip the terminal point.

In this text we will denote vectors in boldface type such as a, b, v, w, and

x, and we will denote scalars in lowercase italic type such as a, k, v, w,
and x. When we want to indicate that a vector v has initial point A and
terminal point B, then we will write :

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 Equivalent Vectors and
Zero Vectors
Vectors with the same length and direction, such as those in figure
below, are said to be equivalent. Since we want a vector to be
determined solely by its length and direction, equivalent vectors
are regarded to be the same vector even though they may be in
different positions. Equivalent vectors are also said to be equal,
which we indicate by writing v = w.

The vector whose initial and terminal points coincide has length
zero, so we call this the zero vector and denote it by 0. The zero
vector has no natural direction, so we will agree that it can be
assigned any direction that is convenient for the problem at hand.
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 Vector Addition
Parallelogram Rule for Vector Addition
If v and w are vectors in 2-space or 3-space that are positioned so
their initial points coincide, then the two vectors form adjacent sides
of a parallelogram, and the sum is the vector represented by the
arrow from the common initial point of and to the opposite vertex of
the parallelogram.

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 Vector Substraction
In ordinary arithmetic we can write a – b = a + (-b), which
expresses subtraction in terms of addition. There is an analogous
idea in vector arithmetic :
The negative of a vector v, denoted by -v, is the vector that has
the same length as v but is oppositely directed, and the difference
of v from w, denoted by w – v, is taken to be the sum
w – v = w + (-v).

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 Scalar Multiplication
If is a nonzero vector in 2-space or 3-space, and if k is a nonzero
scalar, then we define the scalar product of v by k to be the
vector whose length is |k| times the length of v and whose
direction is the same as that of v if k is positive and opposite to
that of v if k is negative. If k=0 or v=0, then we define k.v to be 0.

Figure below shows the geometric relationship between a vector v

and some of its scalar multiples. In particular, observe that (-1).v
has the same length as v but is oppositely directed; therefore,
(-1).v = -v

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 Parallel and Collinear Vectors

Suppose that v and w are vectors in 2-space or 3-space with a common

initial point. If one of the vectors is a scalar multiple of the other, then the
vectors lie on a common line, so it is reasonable to say that they are collinear
(see figure below). However, if we translate one of the vectors, then the
vectors are parallel but no longer collinear. This creates a linguistic problem
because translating a vector does not change it. The only way to resolve this
problem is to agree that the terms parallel and collinear mean the same thing
when applied to vectors. Although the vector 0 has no clearly defined
direction, we will regard it to be parallel to all vectors when convenient.

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 Sum of Three Vectors
Vector addition satisfies the associative law for addition,
meaning that when we add three vectors, say u, v, and w, it does
not matter which two we add first; that is,
u + (v + w) = (u + v) + w
It follows from this that there is no ambiguity in the expression
because the same result is obtained no matter how the vectors are
grouped.

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 Vectors in Coordinate Systems

If a vector v in 2-space or 3-space is positioned with its initial point

at the origin of a rectangular coordinate system, then the vector is
completely determined by the coordinates of its terminal point. We
call these coordinates the components of v relative to the
coordinate system. We will write v = (v1 ,v2) to denote a vector v in
2-space with components (v1, v2), and v = (v1, v2, v3) to denote a
vector in 3-space with components (v1, v2, v3).

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 Vectors Whose Initial Points Is
Not at the Origin
If P1 P2 denotes the vector in 2-space with initial point P1(x1,y1)
and terminal point P2(x2,y2), then the components of this vector
are given by the formula

Moreover, the components of a vector in

3-space that has initial point P1(x1,y1,z1)
and terminal point P2(x2,y2,z2) are given by

Example :

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Vector Operations in Rn
 Vectors in n-Space
Definition
If n is a positive integer, then an ordered n-tuple is a sequence of n
real numbers (v1,v2,...,vn). The set of all ordered n-tuples is called n-
space and is denoted by Rn.

We will denote a vector v in Rn using the notation v = (v1,v2,...,vn) and

we will call 0 = (0,0,...,0) the zero vector.

Definition
Vectors v = (v1,v2,...,vn) and w = (w1,w2,...,wn) in Rn are said to be
equivalent (also called equal) if
v1 = w1, v2 = w2, ..., vn = wn.
We indicate this by writing v = w.

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 Vector Addition and
Scalar Multiplication
Definition
If v = (v1,v2,...,vn) and w = (w1,w2,...,wn) are vectors in Rn, and if k
is any scalar, then we define these operations :
v + w = (v1+w1, v2+w2, ..., vn+wn)
k.v = (k.v1, k.v2, ..., k.vn)
-v = (-v1, -v2, ..., -vn)
w – v = (w1-v1, w2-v2, ..., wn-vn)

Example
If v = (1,-3,2) and w = (4,2,1), then
v + w = (5,-1,3), 2v = (2,-6,4),
-w = (-4,-2,-1), v - w = (-3,-5,1).

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 Illustration in 2-Space
Geometric interpretations of vector addition and scalar multiplication
in 2-Space (R2) are shown in figure below :

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 Properties of Vector Addition
and Scalar Multiplication
Theorem

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 Properties of Vector Addition
and Scalar Multiplication
Theorem

Above theorem can be proved easily using some parts of the

previous theorem. Be aware to distinguish 0 (scalar) and 0 (zero
vector).

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 Norm of a Vector
We will denote the length of a vector v by the symbol ||v||, which is
read as the norm of v, the length of v, or the magnitude of v (the
term “norm” being a common mathematical synonym for length). In
general n-Space, the norm of a vector is defined by “extending”
the Theorem of Pythagoras.
Definition

Example
Norm of the vector v = (-3,2,1) in R3 is

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 Illustration
These figures illustrate why the Theorem of Pythagoras and its
“extension” are suitable to define norm of a vector in R2 and R3.

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 Properties of Norm
There are familiar facts about vectors :
• Distances are nonnegative.
• The zero vector is the only vector of length zero.
• Multiplying a vector by a scalar multiplies its length by the
absolute value of that scalar.
Therefore, we can generalize those facts in this theorem.

Theorem

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 Unit Vectors and
Standard Unit Vectors
A vector of norm 1 is called a unit vector. When a rectangular
coordinate system is introduced in R2 or R3, the unit vectors in the
positive directions of the coordinate axes are called the standard
unit vectors. In R2 these vectors are denoted by
i = (1,0) and j = (0,1)
and in R3 by
i = (1,0,0), j = (0,1,0) and k = (0,0,1)

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 Distance between Two Points
Definition
If P1 and P2 are points in R2 or R3, then the length of the vector P1 P2
is equal to the distance d between the two points. Specifically, if
P1(x1,y1) and P2(x2,y2) are points in R2, then

This is the familiar distance formula from analytic geometry.

Similarly, the distance between the points P1(x1,y1,z1) and
P2(x2,y2,z2) in 3-space is

Those formulas can also be generalized in Rn, for any natural

number n.

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 Illustration and Example

Example

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 Dot Product
Definition

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 Dot Product (Cont.)
For computational purposes it is desirable to have a formula that
expresses the dot product of two vectors in terms of components.
Definition
If u = (u1,u2) and v = (v1,v2) are vectors in R2, then :
u.v = u1.v1 + u2.v2
If u = (u1,u2,u3) and v = (v1,v2,v3) are vectors in R3, then :
u.v = u1.v1 + u2.v2 + u3.v3
In general, if u = (u1,u2,...,un) and v = (v1,v2,...,vn) are vectors in Rn,
then the dot product (also called the Euclidean inner product)
of u and v is denoted by u.v and is defined by
u.v = u1.v1 + u2.v2 + ... + un.vn
Example
Given two vectors in R4 : u = (-1,3,5,7) and v = (-3,-4,1,0), then
u.v = (-1).(-3) + 3.(-4) + (5).(1) + (7).(0) = -4.

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 Angles between Two Vectors
Note that the angle  between u and v is defined as the smallest
angle, that is 0 ≤  ≤ .

From the formula , and since 0 ≤  ≤ , it follows

from properties of cosine function studied in trigonometry that:
•  is acute if u.v > 0
•  is obtuse if u.v < 0
•  = /2 acute if u.v = 0

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 Orthogonal Vectors
Recall from the previous slide that if θ is the angle between two
nonzero vectors u and v in Rn, then  = /2 = 900 if and only if u.v = 0.
Thus, we make the following definition about orthogonality.

Definition
Two nonzero vectors u and v in Rn are said to be orthogonal (or
perpendicular) if u.v = 0. We will also agree that the zero vector in Rn
is orthogonal to every vector in Rn.
A nonempty set of vectors in Rn is called an orthogonal set if all pairs
of distinct vectors in the set are orthogonal. An orthogonal set of unit
vectors is called an orthonormal set.

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 Example
Find the angle between a diagonal of a cube and one of its edges.
Solution Let k be the length of an edge and introduce a
coordinate system as shown in figure below. If we let u1=(k,0,0),
u2=(0,k,0) and u3=(0,0,k), then the vector d=(k,k,k)=u1+u2+u3 is a
diagonal of the cube. It follows from first definition of dot product
that the angle θ between d and the edge u1 satisfies

With the help of a calculator we obtain .

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 Algebraic Properties of
the Dot Product
Note that in special case u = v we have .
Theorem

Theorem

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Vector Projections
 Orthogonal Projection
In many applications it is necessary to “decompose” a vector u into a
sum of two terms, one term being a scalar multiple of a specified
nonzero vector a and the other term being orthogonal to a. For
example, if u and a are vectors in R2 that are positioned so their initial
points coincide at a point Q, then we can create such a decomposition
as follows.
• Drop a perpendicular from the tip of u to the line through a.
• Construct the vector w1 from Q to the foot of the perpendicular.
• Construct the vector w2 = u - w1.

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 Projection Theorem
Since w1 + w2 = w1 + (u – w1) = u, we have decomposed u into a sum
of two orthogonal vectors, the first term being a scalar multiple of a and
the second being orthogonal to a.
Theorem
If u and a are vectors in Rn and if a ≠ 0, then u can be expressed in
exactly one way in the form u = w1 + w2, where w1 is a scalar multiple
of a and w2 is orthogonal to a.
The vector w1 is called the orthogonal projection of u on a or
sometimes the vector component of u along a, and the vector w2 is
called the vector component of u orthogonal to a. Their formula are :

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 Example

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 Linear Combinations,
Linear Independency of Vectors,
and Basis
 Linear Combinations of Vectors

Addition, subtraction, and scalar multiplication are frequently

used in combination to form new vectors. For example, if v1, v2,
and v3 are vectors in Rn, then the vectors
u = 2v1 + 3v2 + v3 and w = 7v1 – 6v2 + 8v3
are formed in this way. In general, we make the following
definition.

Definition
If w is a vector in Rn, then w is said to be a linear combination
of the vectors v1, v2, ..., vr in Rn if it can be expressed in the form
w = k1v1 + k2v2 + ... + krvr
where k1, k2, ..., kr are scalars. These scalars are called the
coefficients of the linear combination.

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 Linear Independence and
Dependence
Definition

In other words, a set of vectors S is a linearly independent if and

only if linear combination of vectors in S is equal to zero vector
implies all of the coefficients are zero.

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 Example

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 Example (Cont.)

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 Span
A set of vector S in Rn is said to spans Rn if every vector of
Rn is a linear combination of vectors in S.
Example:
 Basis of a vector space
A set of vector S in Rn is said to be a basis of Rn if
(i) S is linear independent, and
(ii) S spans Rn.
 Dimension of a vector space
A set of vector S in Rn is said to be a basis of Rn if
(i) S is linear independent, and
(ii) S spans Rn.

The numbers of elements of any basis of Rn are same,

called the dimension of Rn.
Cross product in R3
 Cross Product
We have defined the dot product of two vectors u and v in n-
space. That operation produced a scalar as its result. We will now
define a type of vector multiplication that produces a vector as the
result but which is applicable only to vectors in 3-space.

Definition

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 Cross Product (Cont.)
It is also worth (and it will be easier to remember) noting that a
cross product can be represented in the determinant form :

where i, j and k are the standard unit vectors in R3.

Example
Find u x v, where u = (1,2,-2) and v = (3,0,1).

or equivalently

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 Relationships Involving Cross
Product and Dot Product
Theorem

Note that part (a) and (b) of the theorem mean that u x v is
orthogonal (or perpendicular) to both u and v.
Example
In previous example, u = (1,2,-2), v = (3,0,1) and we obtain
u x v = (2,-7,-6). Since u.(u x v) = (1)(2) + (2)(-7) + (-2)(-6) = 0 and
v.(u x v) = (3)(2) + (0)(-7) + (1)(-6) = 0, u x v is orthogonal to both
u and v, as guaranteed by above theorem.

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 Properties of Cross Product
Theorem

Also note that the cross product is not associative, i.e.

(u x v) x w ≠ u x (v x w).
Therefore we cannot well-define the form u x v x w and need to
make sure which operation should be done first.
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 Geometric Interpretation
of Cross Product
We have known that u x v is orthogonal to both u and v. If u and v
are nonzero vectors, it can be shown that the direction of u x v can
be determined using the following “right-hand rule”:
Let θ be the angle between u and v, and suppose u is rotated
through the angle θ until it coincides with v. If the fingers of the
right hand are cupped so that they point in the direction of rotation,
then the thumb indicates (roughly) the direction of u x v.
Also, from Lagrange’s identity, we can obtain that

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Line Equations in 2-Space and 3-
Space
 Vector and Parametric Equations
of Lines in R2 and R3
A unique line in R2 or R3 is determined by a point x0 on the line
and a nonzero vector v parallel to the line.

Let us begin by deriving an equation for the line L that contains

the point x0 and is parallel to v. If x is a general point on such a
line, then, as illustrated in above figure, the vector x - x0 will be
some scalar multiple of v, say
x – x0 = tv or equivalently x = x0 + tv
As the variable t (called a parameter) varies from -∞ to ∞, the
point x traces out the line L.
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 Example
(a) Find a vector equation and parametric equations of the line in
R2 that passes through the origin and is parallel to the vector
v=(-2,3).
(b) Find a vector equation and parametric equations of the line in
R3 that passes through the point P0(1,2,-3) and is parallel to
the vector v=(4,5-,1).
(c) Use the vector equation obtained in part (b) to find two points
on the line that are different from P0.
Solution
(a) In this case x0=0, so a vector equation of the line is x = t v. If
we let x = (x,y), then this equation can be expressed in vector
form as : (x,y) = t (-2,3).
Equating corresponding components on the two sides of this
equation yields the parametric equations: x = -2t, y = 3t.
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 Example (Cont.)
(b) A vector equation of the line is x = x0 + tv. If we let x = (x,y,z),
and if we take x0 = (1,2,-3), then this equation can be expressed
in vector form as
(x,y,z) = (1,2,-3) + t (4,-5,1)
Equating corresponding components on the two sides of this
equation yields the parametric equations
x = 1 + 4t, y = 2 – 5t, z = -3 + t
(c) A point on the line represented by equation in part (b) can be
obtained by substituting a specific numerical value for the
parameter t . However, since t = 0 produces (x,y,z) = (1,2,-3),
which is the point P0, this value of t does not serve our purpose.
Taking t = 1 produces the point (5,-3,-2) and taking t = -1
produces the point (-3,7,-4). Any other distinct values for t
(except t = 0) would work just as well.

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 Lines Through Two Points in Rn

If x0 and x1 are distinct points in Rn, then the line determined by

these points is parallel to the vector v = x1 – x0, so the line can be
expressed in vector form as
x = x0 + t(x1 – x0)
or, equivalently, as
x = (1-t) x0 + t x1
These are called the two-point vector equations of a line in Rn.

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 Example
Find vector and parametric equations for the line in R2 that passes
through the points P(0,7) and Q(5,0).
Solution
We will see below that it does not matter which point we take to be
x0 and which we take to be x1, so let us choose x0 = (0,7) and
x1 = (5,0). It follows that x1 – x0 = (5,-7) and hence that
(x,y) = (0,7) + t (5,-7)
which we can rewrite in parametric form as
x = 5t, y = 7-7t
Had we reversed our choices and taken x0 = (5,0) and x1 = (0,7),
then the resulting vector equation would have been
(x,y) = (5,0) + t (-5,7)
and the parametric equations would have been
x = 5 – 5t, y = 7t

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 Example (Cont.)

Although those two results look different, they both represent the line
whose equation in rectangular coordinates is
7x + 5y =35
This can be seen by eliminating the parameter t from the parametric
equations (verify).

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Plane Equations in R3
 Plane Determined by
Points and Normals
One learns in analytic geometry that a plane in R3 is determined
uniquely by its “inclination” and one of its points. One way of
specifying slope and inclination is to use a nonzero vector n, called a
normal, that is orthogonal to the line or plane in question. For
example, figure in next slide shows the plane through the point
P0(x0,y0,z0) that has normal n = (a,b,c). The plane are represented by
the vector equation

where P is an arbitrary point (x,y,z) in the plane. The vector P0 P can be

expressed in terms of components as

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 Plane Determined by
Points and Normals (Cont.)
Therefore the equation

is called the point-normal equation of the plane.

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 Example and Further Theorem
The plane in R3 through the point (3,0,7) with n = (4,2,-5) is :
4 (x-3) + 2y - 5(z-7) =0,
or equivalently : 4x +2y – 5z +23 = 0.

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 A Further Theorem
We have learned that we can determine the equation of a plane from
its normal vector.
Conversely, we can determine the normal vector of a plane from given
its equation, as explained in this theorem :

Theorem
If a, b, and c are constants that are not all zero, then an equation of
the form
ax + by + cz + d = 0
represents a plane in R3 with normal n = (a,b,c).

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 Vector and Parametric
Equations of Planes in R3
A unique plane in R3 is determined by a point in the plane and two
noncollinear vectors v1 and v2 parallel to the plane. The best way to
visualize this is to translate the vectors so their initial points are at x0.

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 Vector and Parametric
Equations of Planes in R3 (Cont.)
We will derive an equation for the plane W that contains the point x0
and is parallel to the noncollinear vectors v1 and v2. As shown in
figure below, if x is any point in the plane, then by forming suitable
scalar multiples of v1 and v2, say t1v1 and t2v2, we can create a
parallelogram with diagonal x – x0 and adjacent sides t1v1 and t2v2.
Thus, we have
x – x0 = t1v1 + t2v2 or equivalently x = x0 + t1v1 + t2v2
As the variables t1 and t2 (called parameters) vary independently from
-∞ to ∞, the point x varies over the entire plane W.

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 Example
Find vector and parametric equations of the plane x – y + 2z = 5.
Solution
We will find the parametric equations first. We can do this by solving
the equation for any one of the variables in terms of the other two and
then using those two variables as parameters. For example, solving
for x in terms of y and z yields : x = 5 + y – 2z,
and then using y and z as parameters t1 and t2, respectively, yields the
parametric equations
x = 5 + t1 – 2t2, y = t1, z = t2
To obtain a vector equation of the plane we rewrite these parametric
equations as
(x,y,z) = (5 + t1 – 2t2, t1, t2)
or, equivalently, as
(x,y,z) = (5,0,0) + t1 (1,1,0) + t2 (-2,0,1).
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 Equation of a Plane Through
Three Noncollinear Points
We can also determine a unique plane through three noncollinear
points. The idea is by combining cross product and the point-normal
equation. These are the steps :
1. Given three noncollinear points in R3 : A, B and C. It is
geometrically clear that the vectors AB, AC , BA, BC , CA, CB are
all lie on the plane through A, B and C. Choose two of those
vectors.
2. Compute the cross product between two vectors chosen in step 1.
From properties of cross product, the result will be orthogonal to
both of chosen vectors, and hence orthogonal to the plane.
Therefore, it will be the normal vector of the plane.
3. After we obtain the normal vector, just use the point-normal
equation formula to determine equation of the plane. We just need
to choose one point among three points A, B or C. All of the
choices will obtain equivalent plane equation.
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Equation of a Plane Through
Three Noncollinear Points
 Distance between a Point
and a Plane
Theorem

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 Example

Find the distance D between the point (1,-4,-3) and the plane
2x – 3y + 6z = -1.
Solution
Since the distance formulas in the theorem require that the
equations of the line and plane be written with zero on the right
side, we first need to rewrite the equation of the plane as
2x – 3y + 6z + 1= 0
from which we obtain

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 Distance between Parallel Plane
Two planes are parallel if their normal vectors are parallel.
Distance between two parallel plane is calculated by taking one
arbitrary point from one plane and then calculating the distance
between that point and the other plane.

Example

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 References
Howard Anton and Anton Kaul. (2019). Elementary Linear
Algebra. 12. Wiley. New Jersey. ISBN: 978-1-119-40677-8
Thank You