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A vector is an object that contains both a magnitude and a direction. Force and velocity are the two examples of vector quantities. Understanding the magnitude of the vector would indicate the strength of the force and similarly, the speed of any object is associated with the velocity. In the following article, the magnitude and direction of vectors are explained.
The picture below shows a vector:
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A vector has magnitude (that is the size) and direction:
The length of the line or the arrow given above shows its magnitude and the arrowhead points in the direction.
Now, we can add two vectors by simply joining them head-to-tail, refer the diagram given below for better understanding:
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And it doesn’t matter in which order the two vectors are added, we get the same result anyway:
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Contents
Notation
A vector can often be written in bold, like a or b.
Subtraction of Vectors:
We can also subtract one vector from another, keeping the two points given below in our mind:
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Firstly, we need to reverse the direction of the vectors we want to subtract, this changes the sign of the vector from positive to negative.
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Secondly we need to add them as usual:
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What is the Magnitude of a Vector?
As we know, that vector can be defined as an object which has both magnitudes as well as it has a direction. Now if we have to find the magnitude of a 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.
Magnitude of a Vector Formula:
The magnitude of a vector formula can be used to calculate the length for any given vector and it can be denoted as |v|, where v denotes a vector. So basically, this quantity is used to define the length between the initial point and the end point of the vector.
\[\mid \vec{v} \mid\] = \[\sqrt{x^{2}+y^{2}}\]
\[\mid \vec{v} \mid\] = \[\sqrt{\left ( x_{2}-x_{1} \right )^{2}+\left ( y_{2}-y_{1} \right )^{2}}\]
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.
Direction of A Vector
The direction of a vector is nothing but it can be defined as the measurement of the angle which is made using the horizontal line. One of the methods to find the direction of any given vector AB is :
Tan α is equal to y/x; endpoint at 0.
Where the variable x denotes the change in horizontal line and the variable y denotes a change in a vertical line.
Or we can write that : tan\[\alpha\] = \[\frac{y_{1}-y_{0}}{x_{1}-x_{0}}\]
Where, the variable (x₀, y₀) is known to be the initial point and (x₁, y₁)is known to be the end point.
We may know a vector’s direction and magnitude, but want its x and y lengths (or we can say vice versa):
Magnitude and Direction
Important points to remember, these points given below will be helpful to solve problems:
The magnitude of a vector is always defined as the length of the vector. The magnitude of a vector is 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 is 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.
What Exactly is a Vector and How is it Different from a Scalar?
A vector is any mathematical quantity that includes both magnitude and direction. This might be a bit confusing to understand, however, it is relatively simple once you get the hang of it.
There are certain quantities in the universe that express different things. Mathematicians over the years have broadly classified mathematical quantities into two categories: scalar and vector quantities.
Scalar quantities are those that have only magnitude. Most of the numbers you would have dealt with in school would have been scalar quantities.
Vector quantities, meanwhile, express both magnitude and direction, so they have two aspects to them. Since they cannot generally be used in the same mathematical equations as scalar quantities, there is a whole different branch of mathematics focused on the algebra of vectors.
Let’s look at some common scalar quantities:
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Time
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Mass
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Volume
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Density
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Energy
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Speed
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Temperature
Now let’s look at some common vector quantities:
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Gravity
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Acceleration
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Force
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Displacement
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Thrust
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Velocity
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Angular Momentum
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Linear Momentum
To find out more about vectors, you can click here.
Studying Magnitude of Vectors with Vedantu
On this page, you will find a detailed explanation and examples of vector quantities and using and finding the magnitude of a Vector.
To study more about vectors, you can use the search bar or navigation menu at the top of the page to look for specific study materials that will aid you in learning about vectors. These resources have been created and compiled by experts in mathematics who have worked carefully with Vedantu to provide excellent quality study materials for free. All of the resources available in PDF format on the website and app are downloadable for free if you have a Vedantu account.
For more in-depth information, you can also book a one-on-one session with a maths teacher who will help you learn more about vectors.
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Source: Math Hello Kitty
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