Magnitude of a Vector Formula

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Magnitude of a vector definition

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

. See the introduction to vectors for more about the magnitude of a vector.

Formulas for the magnitude of vectors in two and three dimensions in terms of their coordinates are derived in this page. For a two-dimensional vector a=(a1,a2), the formula for its magnitude is

Magnitude of a Vector Formula

The magnitude of a vector formula helps to summarize the numeric value for a given vector. A vector has a direction and a magnitude. The individual measures of the vector along the x-axis, y-axis, and z-axis are summarized using this magnitude of a vector formula. It is denoted by  |→v|. The magnitude of a vector is always a positive number or zero, i.e., it cannot be a negative number. Let us understand the magnitude of a vector formula using a few solved examples in the end.

What Is the Magnitude of a Vector Formula?

The magnitude of a vector |→A| is |A|. For a given vector with direction ratios along the x-axis, y-axis, and z-axis, the magnitude of the vector is equal to the square root of the sum of the square of its direction ratios. This can be clearly understood from the below magnitude of a vector formula. 

Examples Using Magnitude of a Vector Formula

Example 1: Using magnitude formula, find the magnitude of the

Answer: Magnitude of the given vector = 5.385

FAQs on Magnitude of a Vector Formula

What Is the Magnitude of a Vector Formula?

The magnitude of a vector formula summarizes the numeric

How To Use the Magnitude of a Vector Formula?

In order to use the magnitude of a vector formula, follow the steps given below

  • Step 1: Check for the given parameters. 
  • Step 2: Put the values in the appropriate formula

What Concept Is Behind the Formula For Calculating the Magnitude of a Vector?

The magnitude of a vector refers to the length or size of the vector. It also determines its direction. The concepts behind these formulas include the Pythagorean theorem and the distance formula, which are used to derive the formula of the magnitude of the vector.

What Is the Magnitude of Vector Formula In Words?

For a given vector with direction ratios along the x-axis, y-axis, and z-axis, the magnitude of the vector is equal to the square root of the sum of the square of its direction ratios. 

Magnitude and Direction of Vectors

Magnitude of a Vector

If the coordinates of the initial point and the end point of a vector is given, the Distance Formula can be used to find its magnitude.

What is the Magnitude of a Vector?

As we know, the vector is an object which has both the magnitude as well as direction. To find the magnitude of a vector, we need to calculate the length of the vector. Quantities such as velocity, displacement, force, momentum, etc. are vector quantities. But speed, mass, distance, volume, temperature, etc. are scalar quantities. The scalar has the only magnitude, whereas the vectors have both magnitude and direction.

The magnitude of a vector formula is used to calculate the length for a given vector (say v) and is denoted as |v|. So basically, this quantity is the length between the initial point and endpoint of the vector. To calculate the magnitude of the vector, we use the distance formula, which we will discuss here.

Magnitude of a Vector Formula

Suppose, AB is a vector quantity that has magnitude and direction both. To calculate the magnitude of the vector

Direction of a vector

The direction of a vector is nothing but the measurement of the angle which is made with the horizontal line. One

Solved Questions

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

Answer)We know the Magnitude of a vector formula,

 |b| = (√3²+4²) = √9+16 = √25= 5

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

Answer) We know the Magnitude of a vector formula,

|a| =  (√6²+8²) = √36+64 = √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 = √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:

  • Time
  • Mass
  • Volume
  • Density
  • Energy
  • Speed
  • Temperature

Now let’s look at some common vector quantities:

  • Gravity
  • Acceleration
  • Force
  • Displacement
  • Thrust
  • Velocity
  • Angular Momentum
  • Linear Momentum

FAQs (Frequently Asked Questions)

1. What is the Unit Vector Formula?

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

In simple terms, the unit vector formula is used to find the unit vector of a given vector. To find the unit vector, the given vector is divided by the magnitude of the vector.

Unit vectors have the same components as the respectively given vectors. The only difference is that they have a magnitude of 1.

2. What is the Unit Vector Used for?

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.

Although vectors have both magnitude and direction, sometimes, you might only need to know the direction. So normalising the vector into a unit vector with a magnitude of 1 allows you to focus on the direction of the vector. This is useful because unit vectors have all of the same components as given vectors, the only difference is the magnitude of a unit vector is 1.

3. What is a Vector?

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.

To give you an example, let’s look at two different options: mass and gravity. Both of these are interconnected and affect each other, yet one is a scalar quantity and one is a vector quantity.

The scalar quantity in this example is Mass. This is because Mass has magnitude, but no direction. The vector quantity in this example is Gravity because gravity has both magnitude and direction. So any number or quantity involving gravity would be a vector quantity, while a number denoting mass would be a scalar quantity.

4. What is a Scalar? Give an Example.

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.

Let’s take a look at some common scalars and see why they fit into this category.

Time: Time does not have any direction, just magnitude. If we say “4 hours,” there is no director attached to the number.

Mass: Mass is another quantity that does not involve direction. For example, if we say that the Earth has a mass of 5.973 × 10^24, there is no direction implied.

Now let’s look at some vector quantities and see how they fit into this category.

Acceleration: Any number given for Acceleration will either specify or imply a direction. For example, if you say that a car is accelerating at a particular rate, that implies that the car is moving in a particular direction. Positive acceleration indicates that the object is moving in the same direction as the acceleration.

Gravity: Gravity is another quantity that has direction implied or specified. If you say that Earth has a gravity of 9.8m/s² that implies the downward direction of gravity.

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