What is a Vector?
In Physics terminology, you must have heard about scalar and vector quantities. We often define any physical quantity by magnitude. Hence the physical quantity featured by magnitude is called a scalar quantity. That’s it! But there are also physical quantities that have a certain specific magnitude along with the direction. Such a physical quantity represented by its magnitude and direction is called a vector quantity. Thus, by definition, the vector is a quantity characterized by magnitude and direction. Force, linear momentum, velocity, weight, etc. are typical examples of a vector quantity. Unlike scalar quantity, there is a whole lot to learn about vector quantity.
Before learning about the vector quantities and their properties, let us differentiate between the scalar quantities and vector quantities.
Scalar Quantities
These are the quantities that have only magnitude and no direction.
The scalar quantities are one-dimensional.
With the change in magnitude, scalar quantities also change.
The mathematical operation is done between the two or more scalar quantities results in a scalar quantity.
Simple alphabets are used to denote scalar quantities.
Example - speed, time, mass, volume, etc.
Vector Quantities
These quantities have both magnitude and direction.
They can be one, two or three-dimensional.
The vector quantities change when both magnitude and direction are changed.
The resultant of two or more vector quantities is a vector quantity when mathematical operations are applied.
To denote these quantities, an arrowhead is made above the alphabets.
Example - displacement, force, velocity, acceleration, etc.
Vectors are denoted by an arrow marked over a signifying symbol. For example,
\[\overrightarrow{a}\] or \[ \overrightarrow{b}\] \[ \overrightarrow{b}\]
The magnitude of the vector \[\overrightarrow{a}\] and \[\overrightarrow{b}\] is denoted by ∥a∥ and ∥b∥ , respectively.
Examples of the vector are force, velocity, etc. Let’s see below how it is represented
Velocity vector:
\[\overrightarrow{v}\]
Force vector:
\[\overrightarrow{F}\]
Linear momentum:
\[\overrightarrow{p}\]
Acceleration vector:
\[\overrightarrow{a}\]
Force is a vector because the force is the magnitude of intensity or strength applied in some direction. Velocity is the vector where its speed is the magnitude in which an object moves in a particular path.
Classification Of Vectors
There are various types of vectors that are used in Physics and Mathematics. Beneath are the names and descriptions of these vectors:
Zero Vector - It is the type of vector whose magnitude is equal to zero.
Position Vector - The vector which describes the position of a point in a cartesian system with respect to the origin is known as the position vector.
Unit Vector - The magnitude of this vector is equal to unity.
Like Vectors - Like vectors are the vectors having the same direction.
Unlike Vectors - These are the vectors having opposite directions.
Co-initial Vectors - The co-initial vectors have the same starting point.
Equal Vectors - The vectors which have the same magnitude, as well as the direction, are said to be equal vectors.
Coplanar Vectors - Coplanar vectors are the vectors that are parallel to the same plane or lie in the same plane.
Collinear Vectors - These are the vectors that are parallel to each other irrespective of their magnitudes and direction are known as collinear vectors.
Negative of a Vector - The two vectors having the same magnitude but different directions (opposite direction) are said to be the negative vectors of each other.
Displacement Vector - It is the vector that represents the displacement of a point from one position to another.
Two-Dimensional Vectors Depiction
Two- dimensionally vectors can be represented in two forms, i.e. geometric form, rectangular notation, and polar notation.
1. Geometric Depiction of Vectors
In regular simple words, a line with an arrow is a vector, where the length of the line is the magnitude of a vector, and the arrow points the direction of the vector.
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2. Rectangular Depiction
In this form, the vector is placed on the x and y coordinate system as shown in the image
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The rectangular coordinate notation for this vector is
\[\overrightarrow{v}\] = (6,3). An alternate notation is the use of two-unit vectors î = (1,0) and ĵ = (0,1) so that v = 6î + 3ĵ.
3. Polar Depiction
In the polar notation, we specify the vector magnitude r, r≥0, and angle θ with the positive x-axis.
Now we will read different vector properties detailed below.
Equality of Vectors
If you compare two vectors with the same magnitude and direction are equal vectors. Therefore, if you translate a vector to position without changing its direction or rotating, i.e. parallel translation, a vector does not change the original vector. Both the vectors before and after changing position are equal vectors. Nevertheless, it would be best if you remembered vectors of the same physical quantity should be compared together. For example, it would be practicable to equate the Force vector of 10 N in the positive x-axis and velocity vector of 10 m/s in the positive x-axis.
Vector Addition
Think of two vectors a and b, their sum will be a + b.
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The image displays the sum of two vectors formed by placing the vectors head to tail.
Vector addition follows two laws, i.e. Commutative law and associative law.
A. Commutative Law - the order
in which two vectors are added does not matter. This law is also referred to as parallelogram law. Consider a parallelogram, two adjacent edges denoted by a + b, and another duo of edges denoted by, b + a. Both the sums are equal, and the value is equal to the magnitude of diagonal of the parallelogram
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Image display that parallelogram law that proves the addition of vector is independent of the order of vector, i.e. vector addition is commutative
B. Associative Law - the addition of three vectors is independent of the pair of vectors added first.
(a+b)+c=a+(b+c).
Vector Subtraction
First, understand the vector -a. It is the vector with an equal magnitude of a but in the opposite direction.
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The image shows two vectors in the opposite direction but of equal magnitude.
Therefore, the subtraction of two vectors is defined as the addition of two vectors in the opposite direction.
x - y = x + (-y)
Vector Multiplication by a Scalar Number
Consider a vector \[\overrightarrow{a}\] with magnitude ∥a∥ and a number ‘n’. If a is multiplied by n, then we receive a new vector b. Let us see. Vector \[\overrightarrow{b}\]= n \[\overrightarrow{a}\] The magnitude of the vector \[\overrightarrow{b}\] is ∥na∥.
The direction of the vector \[\overrightarrow{b}\] is the same as that of the vector a \[\overrightarrow{a}\]
If the vector \[\overrightarrow{a}\] is in the positive x-direction, the vector b \[\overrightarrow{b}\] will also point in the same direction, i.e. positive x-direction.
Suppose if we multiply a vector with a negative number n whose value is -1. Vector \[\overrightarrow{b}\] will be in the opposite direction of the vector \[\overrightarrow{a}\]
The Vector Product
The cross product of two vectors is equal to the product of the magnitude of the two given vectors and sine of the angle between these vectors. The vector product is represented as
A x B = |A| |B| sin θ nˆ
Where,
A and B are two vectors
|A| = magnitude of vector A
|B| = magnitude of vector B
θ = angle between the vectors A and B
\[\hat{n}\] = unit vector perpendicular to the plane containing the two vectors
Some properties of the vector product are discussed below:
The cross-product follows the ant-commutative law. This means it does obey the commutative property.
A x B ≠ B x A
But, A x B = (-B) x A
It follows the distributive property.
A x (B + C) = A x B + A x C
When the vectors are perpendicular to each other then the vector product is maximum.
Due to parallel and anti-parallel vectors, the cross product becomes zero.
When a vector gets multiplied by itself, then it results in a zero vector.
The orthogonal unit vectors show the cross product in the following manner,
i x i = j x j = k x k = 0
i x j = k, j x k = i, k x i = j
j x i = -k, k x j = -i, i x k = -j
Fun Facts
Do you know, scalar representation of vector quantities like velocity, weight is speed, and mass, respectively?
Scalar multiplication of vector fulfills many of the features of ordinary arithmetic multiplication like distributive laws
a(x + y) = xa + xb(a + b)y = ay + by 1x = x(−1)x = -x0a = 0
FAQs on Properties of Vector
1. What is a Unit Vector?
Vector whose magnitude is 1 unit. Therefore, a unit vector is majorly used to denote the direction of vector quantities. In Cartesian coordinates, usually: î, ĵ, k̂ = unit vector in x, y, z-direction respectively
The position vector of any object can be signified in Cartesian coordinates as:
r = xî + yĵ + zk̂.
When direction and not magnitude is the major interest for any vector quantity, then vectors are normalized to unit length magnitude. Any vector is the combination of the sum of the unit vector and scalar coefficients. The unit vector in the x-axis, y-axis, z-axis direction is i, j, and k, respectively.
2. What is the Zero Vector?
Zero vector with no direction is an exception to vectors having direction. As the name suggests, the zero vector is a vector of zero magnitudes. Because of its zero magnitudes, the zero vector does not point in any direction. There can only be a single vector of zero magnitudes. It is denoted by 0 as the length or magnitude is zero. Hence we say the zero vector.
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The image displays two vectors with some magnitude pointing in a particular direction, whereas the zero vector is in the form of a simple dot with zero magnitudes and also does not point in any direction on the left and right, respectively.
3. Where will students get an article on the topic of "Properties of Vectors"?
Several course materials are available on the topic of "Properties of Vectors" on the internet. But the best article on this topic is provided by Vedantu. The experts have created the content which discusses the meaning of vectors, representation of vectors and types of vectors. All these concepts are explained in detail so that students can grasp them easily. Other study materials like the NCERT Solutions and important questions on this topic are also present on the official website of Vedantu. Students can even download the Vedantu app to get these educational materials.
4. Discuss some examples of vector quantities.
Vector quantities are known to possess both magnitude and direction.
Some examples of these quantities are mentioned below:
Force - A boy exerted 30 N force in the east direction.
Displacement - A car moves 60 km in the north direction.
Velocity - The velocity of the freely falling body is 12 meters per sec.
Acceleration - Acceleration due to gravity is 9.8 meters per second square (downwards).
Momentum - The momentum of a bus is 300 kg meters per second (South east direction).
5. What is the meaning of the vector product?
The vector product is also known as the cross product. It is defined as the product of two vectors that result in a vector quantity. This resultant vector is perpendicular to the given two vectors and is normal to the plane in which the two vectors are present. To calculate this product, the angle between the two given vectors is used. The direction of the resultant vector can be determined with the help of the Right-hand rule. The commutative law is not followed by the cross-product.
It obeys the distributive and anti-commutative properties.