Vector:
A vector is a quantity that has both magnitude and direction. (Magnitude just
means 'size'.)
Examples of Vector Quantities:


I travel 30 km in a Northerly direction (magnitude is 30
North - this is a displacement vector)
The easterly wind is 3m/sec.
km, direction is
Other examples of vectors include:
Acceleration, momentum, queue, angular momentum, magnetic and electric
fields
Each of the examples above involves magnitude and direction.
Note: A vector is not the same as a scalar. Scalars have magnitude only. For
example, a weight of 50kg is a scalar quantity, since no direction is given. Other
examples of scalar quantities are:
Volume, density, temperature, mass, speed, time, length, distance, work and
energy.
Each of these quantities has magnitude only, and do not involve direction.
We will use a bold capital letter to name vectors. For example, a force vector
could be written as F.
Alternative vector notations


Some textbooks write vectors using an arrow above the vector name, like
this:
You will also see vectors written using matrix-like notation. For example,
the vector acting from (0, 0) in the direction of the point (2, 3) can be
written [2 3]
A vector is drawn using an arrow. The length of the arrow indicates the
magnitude of the vector. The direction of the vector is represented by (not
surprisingly :-) the direction of the arrow.
Example 1 - Vectors
The displacement vector A has direction 'up' and a magnitude of 4 cm.
Vector B has the same direction as A, and has half the magnitude (2 cm).
Vector C has the same magnitude as A (4 units), but it has different direction.
Vector D is equivalent to vector A. It has the same magnitude and the same
direction. It doesn't matter that A is in a different position to D - they are still
considered to be equivalent vectors because they have the same magnitude
and same direction. We can write:
A=D
Note: We cannot write A = C because even though A and C have the same
magnitude (4 cm), they have different direction. They are not equivalent.
Free and Localized Vectors
So far we have seen examples of "free" vectors. We draw them without any fixed
position.
Another way of representing vectors is to use directed line segments. This
means the vector is named using an initial point and a terminal point. Such a
vector is called a "localized vector".
Example 2 - Localized Vectors
A vector OP has initial point O and terminal point P. When using directed line
segments, we still use an arrow for the drawing, with P at the arrow end. The
length of the line OP is an indication of the magnitude of the vector.
We could have another vector RS as follows. It has initial point R and terminal
point S.
Because the 2 vectors have the same magnitude and the same direction (they
are both horizontal and pointing to the right), then we say they are equal. We
would write:
OP = RS
Note that we can move vectors around in space and as long as they have the
same vector magnitude and the same direction, then they are considered equal
vectors.
Magnitude of a Vector
We indicate the magnitude of a vector using vertical lines on either side of the
vector name.
The magnitude of vector PQ is written |PQ|.
We also used vertical lines like this earlier in the Numbers chapter (where it was
called 'absolute value', a similar concept to magnitude).
So for example, vector A above has magnitude 4 units. We would write the
magnitude of vector A as:
| A | =4
Scalar Quantities
A scalar quantity has magnitude, but not direction.
For example, a pen may have length "10 cm". The length 10 cm is a scalar
quantity - it has magnitude, but no direction is involved.
Scalar Multiplication
We can increase or decrease the magnitude of a vector by multiplying the vector
by a scalar.
Example 3 - Scalar Multiplication
In the examples we saw earlier, vector B (2 units) is half the size of vector A
(which is 4 units) . We can write:
B = 0.5 A
This is an example of a scalar multiple. We have multiplied the vector A by the
scalar 0.5.
Example 4 - Scalar Multiplication
Vectors in Opposite Directions
We have 2 teams playing a tug-of-war match. At the beginning of the game, they
are very evenly matched and are pulling with equal force in opposite directions.
We could name the vectors OA and OB.
Zero Vectors
A zero vector has magnitude of 0. It can have any direction.
A vector may have zero magnitude at an instance in time. For example, a boat
bobbing up and down in the water will have a positive velocity vector when
moving up, and a negative velocity vector when moving down. At the instant
when it is at the top of its motion, the magnitude is zero.
In the tug-of-war example above, the teams are evenly matched at a certain
instant and neither side is able to move. In this case, we would have:
OA + OB = 0
The 2 force vectors OA and OB, operating in opposite directions, cancel each
other out.
Unit Vectors
A unit vector has length 1 unit and can take any direction.
A one-dimensional unit vector is usually written i.
Example 5 - Unit Vector
In the following diagram, we see the unit vector (in red, labeled i) and two other
vectors that have been obtained from i using scalar multiplication (2i and 7i).
Vectors in 2 Dimensions
On this page...
Components of Vectors
Magnitude of a Vector
Direction of a Vector
So far we have considered 1-dimensional vectors only.
Now we extend the concept to vectors in 2-dimensions. We can use the familiar
x-y coordinate plane to draw our 2-dimensional vectors.
The vector V shown above is a 2-dimensional vector drawn on the x-y plane.
Components of Vectors
Reading from the diagram above, the x-component of the vector V is 6 units.
The y-component of the vector V is 3 units.
We can write these vector components using subscripts as follows:
Vx = 6 units
Vy = 3 units
Magnitude of a 2-dimensional Vector
The magnitude of a vector is simply the length of the vector. We can use
Pythagoras' Theorem to find the length of the vector V above.
Recall (from Section 1, Vector Concepts) that we write the magnitude of V using
the vertical lines notation | V |.
We have:
Magnitude of V =|V|
=root(6 ^2 +3^ 2)
=root(45)
=6.71 units
Direction of a 2-dimensional Vector
To describe the direction of the vector, we normally use degrees (or radians)
from the horizontal, in an anti-clockwise direction.
We use simple trigonometry to find the angle. In the above example, we know
the opposite (3 units) and the adjacent (6 units) values for the angle (θ) we
need.
So we have:
tan θ =3 6 =0.5
This gives:
θ = arctan 0.5
= 26.6°
(= 0.464 radians)
So our vector has magnitude 6.71 units and direction 26.6° up from the right
horizontal axis.
4. Adding Vectors (in 2 dimensions)
Let's first have a play. The following Flash interactive involves a Cessna that is
trying to land on the runway, but it is a bit windy.
You are the pilot. If you have lined up the Cessna properly, you'll be able to land.
If not, you need to go around and try again. You can only land towards the north
(towards the top).
Choose your wind direction and away you go. You can steer with the right and
left arrow keys on your keyboard.
Have a fly around and then attempt your landing. The challenge is to put it down
exactly in the center of the runway.
Adding Vectors Using a Parallelogram
In the Flash interactive activity above, you would have noticed a parallelogram of
forces that changed with the change in heading of the plane or the wind direction.
The parallelogram is an alternative method to using triangles. If we add the the
blue (heading) vector and the black (wind) vector the resultant vector is the red
ground direction vector. In the image, the ground direction is due North.
Unit Vectors and Components of a Vector (2-D)
We met the idea of a "unit vector" before in 1. Vector Concepts. We now extend
the idea for 2-dimensional vectors.
The diagram shows a unit vector in the x-direction (called vector i) and another in
the y-direction (called vector j).
We can write any 2-dimensional vector in terms of the unit vectors i and j.
Example
In an earlier example, we had the following vector:
We could write the components of the vector V as follows.
Vx = 6 i
Vy = 3 j
So we can write the vector V using unit vectors as follows:
V=6 i+3 j
Dot Product (aka Scalar Product) in 2
Dimensions
Also on this page:
If we have any 2 vectors P and Q, the dot product of P and Q is given by:
P ⋅ Q = |P| |Q| cos θ
where
|P| and |Q| are the magnitudes of P and Q respectively, and
θ is the angle between the 2 vectors.
The dot product of the vectors P and Q is also known as the scalar product
since it always returns a scalar value.
The term dot product is used here because of the • notation used and because
the term "scalar product" is too similar to the term "scalar multiplication" that we
learned about earlier.
Example 1
a. Find the dot product of the force vectors F1
to each other as in the diagram.
= 4 N and F2 = 6 N acting at 40°
b. Find the dot product of the vectors P and Q if |P| = 7
units and they are acting at right angles to each other.
units and |Q| = 5
The second example illustrates an important point about how scalar products can
be used to find out if vectors are acting at right angles, as follows.
Dot Product and Perpendicular Vectors
If 2 vectors act perpendicular to each other, the dot product (ie scalar product)
of the 2 vectors has value zero.
This is a useful result when we want to check if 2 vectors are actually acting at
right angles.
Dot Products of Unit Vectors
For the unit vectors i (acting in the x-direction) and j (acting in the y-direction),
we have the following dot (ie scalar) products (since they are perpendicular to
each other):
i⋅ j=j⋅ i=0
Example 2
What is the value of these 2 dot products:
a. i
⋅ i
b. j
⋅ j
Answer
Alternative Form of the Dot Product
Recall that vectors can be written using scalar products of unit vectors.
If we have 2 vectors P and Q defined as:
P=ai+bj
Q = c i + d j,
where
a, b, c, d are constants;
i is the unit vector in the x-direction; and
j is the unit vector in the y-direction,
then it can be shown that the dot product (scalar product) of P and Q is given
by:
P ⋅ Q = ac +
bd
Answer
Example 3 - Alternative Form of the Dot Product
Find P • Q if
P = 6 i + 5 j and
Q=2i−8j
Answer
Now we see another use for the dot product − finding the angle between vectors.
Angle Between Two Vectors
We can use the dot product to find the angle between 2 vectors. For the vectors
P and Q, the dot product is given by
P ⋅ Q = |P| |Q| cos θ
Rearranging this formula we obtain the cosine of the angle between P and Q:
cos θ=P⋅Q/ |P||Q|
To find the angle, we just find the inverse cosine of the expression on the right.
So the angle θ between 2 vectors P and Q is given by
θ=arccos(P⋅Q /|P||Q| )
Example 4
Find the angle between the vectors P
= 3 i − 5 j and Q = 4 i + 6 j
The 3-dimensional Co-ordinate System
We can expand our 2-dimensional (x-y) coordinate system into a 3-dimensional
coordinate system, using x-, y-, and z-axes.
The x-y plane is horizontal in our diagram above and shaded green. It can also
be described using the equation z = 0, since all points on that plane will have 0
for their z-value.
The x-z plane is vertical and shaded pink above. This plane can be described
using the equation y=0
The y-z plane is also vertical and shaded blue. The y-z plane can be described
using the equation x=0 .
We normally use the 'right-hand orientation' for the 3 axes, with the positive xaxis pointing in the direction of the first finger of our right hand, the positive y-axis
pointing in the direction of our second finger and the positive z-axis pointing up in
the direction of our thumb.
7. Vectors in 3-D Space
On this page...
Magnitude of a 3-D Vector
Adding 3-D Vectors
Dot Product of 3-D Vectors
Direction Cosines
Angle Between Vectors
Application
We saw earlier how to represent 2-dimensional vectors on the x-y plane.
Now we extend the idea to represent 3-dimensional vectors using the x-y-z axes.
(See The 3-dimensional Co-ordinate System for background on this).
Example
The vector OP has initial point at the origin O (0, 0,
(2, 3, 5). We can draw the vector OP as follows:
0) and terminal point at P
==================================================
Array: An array is a systematic arrangement of objects, usually in rows and columns.
N-dimensional vector can be expressed by array, and vice versa.
For example, I have your heigh, which I use a variable to denote:
X= [172 180 171 163 163….];
Matrix:
A matrix is a rectangular array of numbers or other mathematical objects, for which
operations such as addition and multiplication are defined.[4] Most commonly, a matrix
over a field F is a rectangular array of scalars from F.[5][6] Most of this article focuses on
real and complex matrices, i.e., matrices whose elements are real numbers or complex
numbers, respectively. More general types of entries are discussed below. For instance,
this is a real matrix:
The numbers, symbols or expressions in the matrix are called its entries or its elements.
The horizontal and vertical lines in a matrix are called rows and columns, respectively.
Size
The size of a matrix is defined by the number of rows and columns that it contains. A
matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, while m
and n are called its dimensions. For example, the matrix A above is a 3 × 2 matrix.
Matrices which have a single row are called row vectors, and those which have a single
column are called column vectors. A matrix which has the same number of rows and
columns is called a square matrix. A matrix with an infinite number of rows or columns
(or both) is called an infinite matrix. In some contexts such as computer algebra programs
it is useful to consider a matrix with no rows or no columns, called an empty matrix.
The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the i, j,
(i,j), or (i,j)th entry of the matrix, and most commonly denoted as ai,j, or ai j. Alternative
notations for that entry are A[i,j] or Ai,j. For example, the (1,3) entry of the following
matrix A is 5 (also denoted a13, a1,3, A[1,3] or A1,3):
Basic operations
There are a number of basic operations that can be applied to modify matrices, called
matrix addition, scalar multiplication, transposition, matrix multiplication, row
operations, and submatrix
Addition, scalar multiplication and transposition
Familiar properties of numbers extend to these operations of matrices: for example,
addition is commutative, i.e., the matrix sum does not depend on the order of the
summands: A + B = B + A. The transpose is compatible with addition and scalar
multiplication, as expressed by (cA)^T = c(A^T) and (A + B)^T = A^T + B^T. Finally,
(A^T)^T = A.
Besides the ordinary matrix multiplication just described, there exist other less frequently
used operations on matrices that can be considered forms of multiplication, such as the
Hadamard product and the Kronecker product. They arise in solving matrix equations
such as the Sylvester equation.
Linear equations
Matrices can be used to compactly write and work with multiple linear equations, i.e.,
systems of linear equations. For example, if A is an m-by-n matrix, x designates a column
vector (i.e., n×1-matrix) of n variables x1, x2, ..., xn, and b is an m×1-column vector, then
the matrix equation
Square matrices
A square matrix is a matrix with the same number of rows and columns. An n-by-n
matrix is known as a square matrix of order n. Any two square matrices of the same order
can be added and multiplied. The entries aii form the main diagonal of a square matrix.
They lie on the imaginary line which runs from the top left corner to the bottom right
corner of the matrix.
It is a square matrix of order n, and also a special kind of diagonal matrix. It is called
identity matrix because multiplication with it leaves a matrix unchanged:
AIn = ImA = A for any m-by-n matrix A.
Symmetric or skew-symmetric matrix[edit source | editbeta]
A square matrix A that is equal to its transpose, i.e., A = AT, is a symmetric matrix. If
instead, A was equal to the negative of its transpose, i.e., A = −AT, then A is a skewsymmetric matrix. In complex matrices, symmetry is often replaced by the concept of
Hermitian matrices, which satisfy A∗ = A, where the star or asterisk denotes the conjugate
transpose of the matrix, i.e., the transpose of the complex conjugate of A.
Invertible matrix and its inverse[edit source | editbeta]
A square matrix A is called invertible or non-singular if there exists a matrix B such that
AB = BA = In.[20][21]
If B exists, it is unique and is called the inverse matrix of A, denoted A−1.
Orthogonal matrix[edit source | editbeta]
An orthogonal matrix is a square matrix with real entries whose columns and rows are
orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix A is orthogonal
if its transpose is equal to its inverse:
which entails
where I is the identity matrix.
An orthogonal matrix A is necessarily invertible (with inverse A−1 = AT), unitary (A−1 =
A*), and normal (A*A = AA*). The determinant of any orthogonal matrix is either +1 or
−1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear
transformation, every orthogonal matrix with determinant +1 is a pure rotation, while
every orthogonal matrix with determinant -1 is either a pure reflection, or a composition
of reflection and rotation.
The complex analogue of an orthogonal matrix is a unitary matrix.
Main operations[edit source | editbeta]
Trace[edit source | editbeta]
The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix
multiplication is not commutative as mentioned above, the trace of the product of two
matrices is independent of the order of the factors:
tr(AB) = tr(BA).
This is immediate from the definition of matrix multiplication:
Also, the trace of a matrix is equal to that of its transpose, i.e.,
tr(A) = tr(AT).
Determinant[edit source | editbeta]
Determinant[edit source | editbeta]
The determinant det(A) or |A| of a square matrix A is a number encoding certain
properties of the matrix. A matrix is invertible if and only if its determinant is nonzero.
The determinant of 2-by-2 matrices is given by
The determinant of 3-by-3 matrices involves 6 terms (rule of Sarrus).
The more lengthy Leibniz formula generalises these two formulae to all dimension
The determinant of a product of square matrices equals the product of their determinants:
det(AB) = det(A) · det(B).
Eigenvalues and eigenvectors
A number λ and a non-zero vector v satisfying
Av = λv
are called an eigenvalue and an eigenvector of A, respectively.[nb 1][30] The number λ is an
eigenvalue of an n×n-matrix A if and only if A−λIn is not invertible, which is equivalent
to
The polynomial pA in an indeterminate X given by evaluation the determinant det(XIn−A)
is called the characteristic polynomial of A. It is a monic polynomial of degree n.
Therefore the polynomial equation pA(λ) = 0 has at most n different solutions, i.e.,
eigenvalues of the matrix.[32] They may be complex even if the entries of A are real.
According to the Cayley–Hamilton theorem, pA(A) = 0, that is, the result of substituting
the matrix itself into its own characteristic polynomial yields the zero matrix.