Magnitude of a Vector

The picture below shows a vector:

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A vector has magnitude (that is the size) and direction:

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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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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:

  • 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.

  • 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:-

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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.


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 α =   \[\frac{y_{1}\: -\: y_{o}}{x_{1}\: -\: x_{0}}\]

Where, the variable (\[x_{0}\] , \[y_{0}\]) is known to be the initial point and  (\[x_{1}\] , \[y_{1}\])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

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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)

x= r × cos(θ)


y= r × sin (θ)

r = \[\sqrt{x^{2} + y^{2}}\]

θ = \[tan^{-1}\] (y/x)

 

Important points to remember, these points given below will be helpful to solve problems: 

  1. The magnitude of a vector is always  defined as the length of the vector. The magnitude of a vector is always denoted as ∥a∥.

  2. For a two-dimensional vector a,  where a = (\[a_{1}\],\[a_{2}\]), ||a|| =  \[\sqrt{a_{1}^{2}\: +\: a_{2}^{2}}\]

  3. For a three-dimensional vector a, where a = (\[a_{1}\],\[a_{2}\],\[a_{3}\]), ||a|| = \[\sqrt{a_{1}^{2}\: +\: a_{2}^{2}\: +\: a_{3}^{2}}\]

  4. 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 = (\[a_{1}\],\[a_{2}\],\[a_{3}\],\[a_{4}\] ), ||a|| = \[\sqrt{a_{1}^{2}\: +\: a_{2}^{2}\: +\: a_{3}^{2}\: +\: a_{4}^{2}}\]

Questions to Be Solved:

Question 1)What is the magnitude of the vector b = (2, 3) ?

Answer)We know the Magnitude of a vector formula,

 |b| = (\[\sqrt{3^{2} + 4^{2}}\]) = \[\sqrt{9 + 16}\] = \[\sqrt{25}\] = 5

Question 2)What is the magnitude of the vector a = (6, 8) ?

Answer) We know the Magnitude of a vector formula,

|a| =  (\[\sqrt{6^{2} + 8^{2}}\]) = \[\sqrt{36 + 64}\] = \[\sqrt{100}\] = 10

Question 3) Find the magnitude of a 3d vector 2i + 3j + 4k.

Answer) We know, the magnitude of a 3d vector xi + yj + zk = \[\sqrt{x^{2} + y^{2} + z^{2}}\]

Therefore, the magnitude of a 3d vector , that is 2i + 3j + 4k is equal to

\[\sqrt{x^{2} + y^{2} + z^{2}}\] = \[\sqrt{(2)^{2} + (3)^{2} + (4)^{2}}\] = 5.38

Hence, the magnitude of a 3d vector given, 2i + 3j + 4k ≈ 5.38.

Note: The symbol ≈ denotes approximation.

FAQ (Frequently Asked Questions)

Question 1. What is the Unit Vector Formula?

Answer. Unit Vector Formula. A unit vector can be defined as a vector that has a magnitude equal to 1. They are labeled with a " ". Suppose any vector can become a unit vector when we divide it by the vector's magnitude. Vectors are basically written in xyz coordinates.

Question 2. What is the Unit Vector Used for?

Answer. Unit Vectors. Vector quantities generally have a direction as well as a magnitude.  In such cases, for convenience, vectors are often referred to as "normalized" to be of unit length. These unit vectors are commonly used to indicate and show direction, with a scalar coefficient providing the magnitude.

Question 3. What is a Vector?

Answer. A vector is known to be a quantity or a phenomenon that has generally two independent properties:

1) Magnitude 

2) Direction

The term vector is also used to denote the geometrical or mathematical representation of such a quantity. A quantity or a phenomenon that exhibits only magnitude, with no specific direction given, is generally referred to as a scalar quantity.

Question 4. What is a Scalar? Give an Example.

Answer. Scalar, a physical quantity that is completely described by its magnitude; examples of scalars are volume, density, speed, energy, mass, and time. Other quantities, like force and velocity,have both magnitude and direction and are known as vectors.