Units Dimensions Vector Formula - Definition, Analysis
All the physical quantities are measured in terms of a standard measurement, known as a unit, for example, mass is measured in kilograms, similarly, length is measured in terms of meters.
Murtaza Mushtaq8 Sept, 2023
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Units, Dimensions Vector Formula are the physical quantities are measured in terms of a standard measurement, known as a unit, for example, mass is measured in kilograms, similarly, length is measured in terms of meters. Dimension is defined as the power to which fundamental or base units must be raised to represent the given quantity. The physical quantities known as vectors have both magnitude and direction like velocity, displacement, etc.
What is a Unit?
In Unit Dimension And Vector there are fundamental and derived quantities, fundamental quantities are those which are independent of other quantities, and units that are used to measure these quantities are called fundamental units. Derived quantities are derived from fundamental quantities and units that measure these quantities are called derived units. International System of Units There are three international systems of units namely CGS, FPS, and MKS. The three base units for length, mass, and time in these systems were as follows: In CGS units are centimeter, gram, and second respectively. In the FPS system, they were foot, pound, and second respectively. In MKS they are meter, kilogram, and second respectively. The system of units which is at present internationally accepted for measurement is the Système Internationale d’ Unites (French for International System of Units), abbreviated as SI. If n 1 and n 2 are the numerical values of a physical quantity corresponding to the units u 1 and u 2 , then n 1 u 1 = n 2 u 2Download PDF Units Dimensions Vector Formula
Fundamental and supplementary physical quantities in the SI system There are seven fundamental units/quantities:- Meter: Meter is defined as the distance traveled by light in a vacuum during a time interval of 1/299, 792, 458 of a second.
- Kilogram: One kilogram is the mass of a platinum-iridium alloy cylinder stored at Serves in Paris' International Bureau of Weights and Measures.
- Second: One second is the length of 9192631770 radiation periods that correspond to the change between the two hyperfine levels of the ground state of cesium-133 atoms.
- Kelvin: Kelvin is defined as 1/273.16 of the triple point of water's thermodynamic temperature.
- Ampere: The current which, when flowing in each of two parallel conductors of infinite length and negligible cross-section and placed one meter apart in vacuum, causes each conductor to experience a force of 2 1 0 -7 newtons per meter of length is known as one ampere.
- Candela : The luminous intensity in the perpendicular direction of a surface of a black body of area 1/600000 m 2 at the temperature of solidifying platinum under a pressure of 101325 N m -2 is known as one candela.
- Mole : The amount of a substance that contains as many elementary particles or entities as there are atoms present in 12g of carbon-12 or C-12 is known as one mole.
| Physical Quantity | SI Unit | Symbol |
| Length | meter | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Thermodynamic temperature | kelvin | K |
| Intensity of light | candela | cd |
| Amount of substance | mole | mol |
| Supplementary Quantity | SI Unit | Symbol |
| Plane Angle | Radian | rad |
| Solid Angle | Steradian | sr |
What are the Dimensions?
Unit Dimension And Vector: Dimensions are the basic units that physicists employ to define the characteristics of the physical universe. Dimension is defined as the power to which fundamental or base units must be raised to represent the given quantity. The dimensions of some fundamental quantities are given as,| Physical Quantity | SI Unit | Dimensional formula |
| Length | m | L 1 |
| Mass | kg | M 1 |
| Time | s | L 1 |
| Electric current | A | A 1 |
| Temperature | K | K 1 |
What is a Dimensional Formula?
The dimensional formula represents how a physical quantity is expressed in terms of the base dimensions (L, M, T, A, K). It is a way to describe the dimensions of a quantity using the symbols of the base dimensions with appropriate powers . It is possible to represent all physical quantities as combinations of these fundamental dimensions increased to certain powers. For example, Speed has dimensions of length and time, L T . So dimensional formula of speed is M 0 L 1 T 1 . Dimensions of some physical quantities| Physical Quantity | SI Unit | Dimensional formula |
| Density | kgm -3 | M 1 L -3 |
| Area | m 2 | L 2 |
| Force | kgms -2 | MLT -2 |
| Energy | kgm 2 s -2 (joule) | ML 2 T -2 |
| Magnetic flux | wb (weber) | ML 2 T -2 A -2 |
| Area | m 2 | L 2 |
| Inertia | kgm 2 | M 1 L 2 T 0 |
| Resistance | kgm 2 s -3 A -2 (ohm) | ML 2 T -3 A -2 |
| Kinetic Energy | kgm 2 s -2 | ML 2 T -2 |
What is Dimensional Analysis?
Unit Dimension And Vector:Dimensional analysis helps us to deduce the relationship between physical quantities with the help of dimensions and units of measurement. Dimensional analysis allows us to check the correctness of a physical equation, whether it is consistent or not. Let us take an example to understand if an equation is consistent or not using dimensional analysis, v = u + at, where v and u are velocities of a body, a is the acceleration of the body, and t is time. Let’s write the dimensions of each quantity, [v] =[u] = [LT -1 ] [a] = [LT -2 ] [t] = [T] Now, [LT -1 ] = [LT -1 ] + [LT -2 ] [T] Thus [LT -1 ] = 2 [LT -1 ] Clearly dimensions of LHS are equal to dimensions of RHS. Thus this equation is consistent as per dimensional analysis.Homogeneity Principle of Dimensional Analysis
It states the dimension of each term of a physical equation must be the same on both sides. If the dimensions of each term are not the same then the equation is not consistent and is wrong. Let's consider an equation, 1/ 2 m v 2 = mgh We will check if this equation is correct or not using the principle of homogeneity of dimensions. Dimensions of LHS are [M] [LT -1 ] 2 = [ML 2 T -2 ] Dimensions of RHS are [M] [LT -2 ] [L] = [ML 2 T -2 ] As dimensions of RHS and LHS are equal, this equation is homogenous and correct as per principle of homogeneity of dimensions.Vectors
Unit Dimension And Vector: Vectors are those physical quantities that have both magnitude and direction and also obey vector algebra. Some examples of vectors are velocity, force, Momentum, and acceleration. Some types of vectors are given as:- Zero or Null Vector: When a vector's magnitude is zero and both its beginning and ending points are the same, the vector is said to be a zero vector.
- Unit Vector: When a vector's magnitude is one unit long, it is referred to as a unit vector.
- Like and Unlike vectors: When two vectors have the same directions, they are said to be like, and when they have opposite directions, they are said to be unlike.
- Collinear Vectors: Collinear vectors, sometimes referred to as parallel vectors, are vectors whose magnitude and direction are in a parallel or the same line.
- Equal Vectors: Even though two vectors have different initial points, they are said to be equal when their direction and magnitude are both equal.
Vector Addition and Subtraction
Vectors are not added algebraically, they are added geometrically. Triangle Law of Vector Addition According to the triangle law of vector addition, the magnitude and direction of the resultant vector are represented on the third side of the triangle when two vectors are represented as two sides of the triangle with the order of magnitude and direction. With A as the vector's tail and B as its head, draw the line AB. Draw a second line BC to represent vector b, with B as the head and C as the tail. Join the line AC now, with A as the head and C as the tail. The resultant sum of the vectors a and b is shown by the line AC. Now the sum of vectors a and b is given by c . => c = a + b The magnitude of c is given as, | c | = a 2 + b 2 + 2abcos where a is the magnitude of vector a, b is the magnitude of vector b and is the angle between vector a and b. Let the resultant make an angle of Φ with vector a, then: tan = bsin a+bcosa
Dot and Cross product of vectors
Dot product also called as a scalar product. The resultant of the dot product is always a scalar quantity. Let a and b be the two vectors and be the angle between them then the dot product of a and b is given by, a.b = | a | | b | cos Here, |a| = magnitude of vector a |b| = magnitude of vector b = angle between the vectors Cross product of two quantities always generates a vector quantity. Let a and b be the two vectors and be the angle between them then the cross product of a and b is given by, a b = | a | | b | sin Here, |a| = magnitude of vector a |b| = magnitude of vector b = angle between the vectorsUnits Dimensions Vector Formula
What is a unit?
A unit is defined as the standard reference for measurements.
What are the seven fundamental physical quantities?
The seven fundamental physical quantities are Length, Mass, Time, Electric current, Temperature, Luminous Intensity, and Amount of substance.
How are two physical quantities with two different units in two different systems related?
As we know, for constant physical quantity
Numerical value ∝ 1/Unit
If n1 and n2 are the numerical values of a physical quantity corresponding to the units u1 and u2, then n1u1 = n2u2
Q4. What is the application of dimensional analysis?
The applications of dimensional analysis are as
To check if a physical equation is consistent or not.
To derive a relationship between physical quantities.
Converting units from one system to another system of units.
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