An
introduction to vectors
Definition
of a vector
A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.
Two vectors are the same if they
have the same magnitude and direction. This means that if we take a vector and
translate it to a new position (without rotating it), then the vector we obtain
at the end of this process is the same vector we had in the beginning.
Two examples of vectors are those
that represent force and velocity. Both force and velocity are in a particular
direction. The magnitude of the vector would indicate the strength of the force
or the speed associated with the velocity.
We denote vectors using boldface as
in a
or b. Especially when writing by hand where one cannot easily
write in boldface, people will sometimes denote vectors using arrows as in a⃗ or b⃗ , or they use other markings. We won't need to use arrows
here. We denote the magnitude of the vector a
by ∥a∥. When we want to refer to a number and stress that it is
not a vector, we can call the number a scalar. We will denote scalars
with italics, as in a or b
.
You can explore the concept of the
magnitude and direction of a vector using the below applet. Note that moving
the vector around doesn't change the vector, as the position of the vector
doesn't affect the magnitude or the direction. But if you stretch or turn the
vector by moving just its head or its tail, the magnitude or direction will change.
(This applet also shows the coordinates of the vector, which you can read about
in another page.)
The magnitude and direction of a
vector. The blue arrow represents a vector a
. The two defining properties of a vector, magnitude and
direction, are illustrated by a red bar and a green arrow, respectively. The
length of the red bar is the magnitude ∥a∥ of the vector a.
The green arrow always has length one, but its direction is the direction of
the vector a. The one exception is when a is the zero vector (the only vector with zero magnitude),
for which the direction is not defined. You can change either end of a by dragging it with your mouse. You can also move a by dragging the middle of the vector; however, changing the
position of the a
in this way does not change the
vector, as its magnitude and direction remain unchanged.
More information about applet.
There is one important exception to
vectors having a direction. The zero vector, denoted by a boldface 0
, is the vector of zero length.
Since it has no length, it is not pointing in any particular direction. There
is only one vector of zero length, so we can speak of the zero vector.
Types of
Vectors
Scalars and Vectors are two rudimentary concepts
in maths and physics. They define the type of quality of any physical or
mathematical object. A scalar quantity is the one that contains only a
magnitude value, whereas a vector quantity is the one that contains a magnitude
as well as a direction. Let us imagine vectors as a line, the measurement of
the line is the magnitude, and the arrow on this line is the direction in which
it is travelling. The classic example of a vector quantity is force. The force
has a magnitude with a direction in which the force acts. In mathematics, a
total of 11 different types of vectors are studied. They are:
- 1. Zero
vector
- 2. Unit
Vector
- 3.
Position Vector
- 4.
Co-initial Vector
- 5. Like
- 6. Unlike
Vectors
- 7.
Co-planar Vector
- 8.
Collinear Vector
- 9. Equal
Vector
- 10.
Displacement Vector
- 11.
Negative Vector
A detailed
description of all the types are given below:
Zero
Vector or Null Vector
When the starting point and the finish point of a
vector coincide with each other, it is known as a zero vector or null vector.
The magnitude of such vectors is zero, and they, in particular, do not
represent any direction.
Unit
Vector
Vectors having a value of exactly one are known as
a unit vector. Unit vectors are very important, and note that if two vectors
are unit vectors, they are not specifically equal. They might have the same
magnitude but can differ in their direction.
Position
Vector
Position vectors are known to determine the
position of any vector. A position vector is nothing but a point on any vector
which tells the position of that vector in a plane
Co-initial
Vectors
Vectors are expressed as co-initial vectors if
they have the same origin point. This implies that the point of origin is
common for these types of vectors, and then they may scatter in different
directions. For example, let us consider two vectors PQ and PR; they are called
co- initial vectors due to the fact they have the same beginning point, i.e.,
P.
Like
Vectors
When two or more vectors share the same direction,
they are known as like vectors.
Unlike
Vectors
When two or more vectors travel in different
directions, they are termed as unlike vectors.
Coplanar
Vectors
Coplanar vectors are vectors (three or more) that
lie in the same plane.
Collinear
Vectors
These are also referred to as parallel vectors
because they lie in the parallel line concerning their magnitude and direction.
Equal
Vectors
Vectors having the same magnitude and the same
directions are known as equal vectors.
Displacement
Vector
The vector KL represents a displacement vector if
a point is moved (displaced) from the position K to L.
Negative
of a Vector
Let us assume that vector K has a magnitude ‘p’
and is in a certain direction, now let us suppose that another vector L is
present having the same magnitude ‘p’ but travels in exactly the opposite
direction of K. Thus, L is referred to as the negative of a vector K. K = -L
Therefore, the negative of any vector is another vector with the same magnitude
but opposite in direction.
Operations
on vectors
We can define a number of operations
on vectors geometrically without reference to any coordinate system. Here we
define addition, subtraction, and multiplication by a scalar.
On separate pages, we discuss two different ways to multiply two vectors
together: the dot product and the cross product.
Addition
of vectors
Given two vectors a
and b, we form their sum a+b, as follows. We translate the vector b until its tail coincides with the head of a. (Recall such translation does not change a vector.) Then,
the directed line segment from the tail of a
to the head of b is the vector a+b
.
The vector addition is the way
forces and velocities combine. For example, if a car is travelling due north at
20 miles per hour and a child in the back seat behind the driver throws an
object at 20 miles per hour toward his sibling who is sitting due east of him,
then the velocity of the object (relative to the ground!) will be in a
north-easterly direction. The velocity vectors form a right triangle, where the
total velocity is the hypotenuse. Therefore, the total speed of the object
(i.e., the magnitude of the velocity vector) is 202+202−−−−−−−−√=202√
miles per hour relative to the
ground.
Addition of vectors satisfies two
important properties.
- The commutative law, which states the order of addition
doesn't matter:
a+b=b+a.
This law is also called the parallelogram law, as
illustrated in the below image. Two of the edges of the parallelogram define a+b, and the other pair of edges define b+a
· . But, both sums are equal to the same
diagonal of the parallelogram.
· The associative law, which states that the
sum of three vectors does not depend on which pair of vectors is added first:
(a+b)+c=a+(b+c).
You can explore the properties of
vector addition with the following applet. (This applet also shows the
coordinates of the vectors, which you can read about in another page.)
The sum of two vectors. The sum a+b
of the vector a
(blue arrow) and the vector b (red arrow) is shown by the green
arrow. As vectors are independent of their starting position, both blue arrows
represent the same vector a and both red arrows represent the
same vector b. The sum a+b
can be formed by placing the tail of the vector b at the head of the vector a.
Equivalently, it can be formed by placing the tail of the vector a at the head of the vector b.
Both constructions together form a parallelogram, with the sum a+b being a diagonal. (For this reason, the commutative law a+b=b+a is sometimes called the parallelogram law.) You can change a and b
by dragging the yellow points.
More information about applet.
Vector
subtraction
Before we define subtraction, we
define the vector −a
, which is the opposite of a.
The vector −a is the vector with the same magnitude as a
but that is pointed in the opposite
direction.
We define subtraction as addition
with the opposite of a vector:
b−a=b+(−a).
This is equivalent to turning vector a around in the applying the above rules for addition. Can
you see how the vector x in the below figure is equal to b−a? Notice how this is the same as stating that a+x=b
, just like with subtraction of
scalar numbers.
Scalar
multiplication
Given a vector a
and a real number (scalar) λ, we can form the vector λa
as follows. If λ is positive, then λa
is the vector whose direction is the same as the direction of a and whose length is λ
times the length of a. In this case, multiplication by λ simply stretches (if λ>1)
or compresses (if 0<λ<1) the vector a
.
If, on the other hand, λ
is negative, then we have to take the opposite of a before stretching or compressing it. In other words, the
vector λa points in the opposite direction of a, and the length of λa
is |λ| times the length of a.
No matter the sign of λ, we observe that the magnitude of λa is |λ| times the magnitude of a:
∥λa∥=|λ|∥a∥
.
Scalar multiplications satisfies
many of the same properties as the usual multiplication.
In the last formula, the zero on the
left is the number 0, while the zero on the right is the vector 0
, which is the unique vector whose
length is zero.
If a=λb
for some
scalar λ,
then we say that the vectors a and b are parallel. If λ is negative, some people say that a and b are anti-parallel, but we will not
use that language.
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