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A Simple Explanation of The Magnitude of a Vector
Magnitude of a Vector – Definition: A vector is composed of magnitude and direction. The magnitude is the size of the vector, and the direction is the angle that the vector makes with the positive x-axis. The magnitude can be measured in any unit of length, such as inches, feet, or meters. Vectors can be added and subtracted, and their magnitudes can be multiplied and divided.
Notation
In mathematics, a function is a set of ordered pairs, where each element in the set corresponds to a unique output. The function assigns a unique output to every input. In other words, for every input there is a unique output. Functions can be represented using notation, which consists of the function’s name followed by parentheses and the input variable. Inside the parentheses, the input is represented by a letter, and the output is represented by a number. For example, the function y = x2 can be represented using notation as y(x).
Subtraction of Vectors:
The subtraction of vectors is the process of finding the difference between two vectors. This can be done by finding the vector sum of the two vectors and then subtracting the second vector from the first.
The subtraction of vectors is a useful tool for finding the location of an object in space. By subtracting the vector for the object’s location from the vector for its destination, you can find the direction and distance between the two points.
What is the Magnitude of a Vector?
The magnitude of a vector is the length of the vector.
Magnitude of a Vector Formula:
The magnitude of a vector is the length of the vector.
Direction of A Vector
The direction of a vector can determined using the right hand rule. The right hand rule states that if you point your right hand in the direction of the vector, your thumb will point in the direction of the vector’s direction.
What is the Magnitude of a Vector?
As we know, that vector can defined as an object which has both magnitudes as well as it has a direction. Now if we have to find the magnitude of vector formula and we need to calculate the length of any given vector. Quantities such as velocity, displacement, force, momentum, etc are the vector quantities. But the quantities like speed, mass, distance, volume, temperature, etc. are known to be scalar quantities. The scalar quantities are the ones that have the only magnitude whereas vectors generally have both magnitude and direction.
Formula for the magnitude of a vector
Note: The magnitude of a vector can never be negative this is because | | converts all the negatives to positive. Thus, we can say that the magnitude of a vector is always positive.
Magnitude and Direction
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Magnitude from Polar Coordinates (r,θ) to Cartesian Coordinates (x,y) |
Magnitude from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ) |
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x= r × cos(θ) y= r × sin (θ) |
r = √x²+y² θ = tan⁻¹ (y/x) |
Important points to remember, these points given below will be helpful to solve problems:
The magnitude of a vector always defined as the length of the vector. The magnitude of a vector always denoted as ∥a∥.
For a two-dimensional vector a, where a = (a₁, a₂ ), ||a|| = √a¹₁+a²₂
For a three-dimensional vector a, where a = (a₁, a₂, a₃), ||a|| = √a²₁+a²₂+a²₃
The formula for the magnitude of a vector always generalized to dimensions that are arbitrary, Now let’s see for example, if we have a four-dimensional vector namely a, where a =a = (a₁, a₂, a₃, a₄), ||a|| = √a²₁+a²₂+a²₃+a²₄
Solved Questions
Q1) What is the magnitude of the vector b = (2, 3) ?
Ans: We know the Magnitude of a vector formula,
|b| = (√3²+4²) = √9+16 = √25= 5
Q2) What is the magnitude of the vector a = (6, 8) ?
Ans: We know the Magnitude of a vector formula,
|a| = (√6²+8²) = √36+64 = √100 = 10
Q3) Find the magnitude of a 3d vector 2i + 3j + 4k.
Ans) We know, the magnitude of a 3d vector xi + yj + zk = √x²+y²+z²
Therefore, the magnitude of a 3d vector , that is 2i + 3j + 4k is equal to
√x²+y²+z² = √(2)²+(3)²+(4)² = 5.38
Hence, the magnitude of a 3d vector given, 2i + 3j + 4k ≈ 5.38.
Note: The symbol ≈ denotes approximation.
