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# Vector Formulas

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Last updated date: 02nd Aug 2024
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## About Vector Formulas

Every object with both a magnitude and a direction is referred to as a vector.

A vector can be drawn geometrically as a guided line section with an arrow representing the direction and a length equal to the magnitude of the vector. From the tail to the head, the vector's orientation is shown. We'll go over the definition of a vector and some vector formulas with examples in this subject. Let's take a look at the idea!

## Vector Formula

### The Concept of Vector Formula

In mathematics, a vector is a representation of an object that includes both magnitude and direction.

If two vectors have the same direction and magnitude, they are the same. This means that if we take a vector and transfer it to a different place, we get a new vector. The vector we get at the end of this phase looks like this, and it's the same vector we had at the start.

In physics, vectors that represent force and velocity are two common examples of vectors. Power and velocity are both acting in the same way. The magnitude of the vector would mean the force's intensity or the velocity's related speed. Since displacement is directly attached to distance, distance and displacement are not the same.

An arrow mark is commonly used to represent a vector.

Also, whose length is proportional to the magnitude and whose direction is the same as the quantity. Scaled vector diagrams with values are often used to describe vector quantities. A displacement vector will be described in the vector diagram.

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## Some Important Definitions and Vector All Formula

### Vector Formula Mathematics

Magnitude

The magnitude of a vector is the length of the vector it is used in the vector formula. The magnitude of the vector a is denoted by |a| For a two-dimensional vector a = (a$_{1}$, a$_{2}$), the formula for its magnitude is

|a| = $\sqrt{a_{1}^{2} + a_{2}^{2}}$

And for three-dimensional vector a = (a$_{1}$, a$_{2}$, a$_{3}$), the formula for its magnitude is

|a| = $\sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$

Direction

A vector's direction is often expressed as a counterclockwise angle of rotation around its "tail" from due East.

A vector with a direction of 30 degrees is a vector that has been rotated 30 degrees, counterclockwise relative, to due east using this convention.

## Vector Formula Physics

### Force

The vector sum of two or more forces is represented by a resultant force, which is a single force.

Like two forces of magnitudes F1 and F2 function on a particle, the effect is as follows:

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### Velocity

The rate of change of an object's direction is represented by a velocity vector.

The magnitude of a velocity vector indicates an object's speed, while the vector direction indicates the object's direction.

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### Triangular Law of Additions

The triangle law of vector addition states that when two vectors are represented as two sides of a triangle of the same order of magnitude and direction, the magnitude and direction of the resulting vector is represented by the third side of the triangle.

As two forces, Vector A and Vector B, function in the same direction, the resulting R is the sum of the two vectors.

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The formula for Triangular law of addition: $\bar{R}$ = $\bar{A}$ + $\bar{B}$

### Parallelogram Law of Addition

When two powers, A Vector B Vector formula, are expressed by the parallelogram's opposite sides, the resultant is represented by the diagonal of a parallelogram taken from the same position.

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The formula for Triangular law of addition: $\bar{R}$ = $\bar{A}$ + $\bar{B}$

### Vector Subtraction

If two powers, Vector A and Vector B, are acting in the opposite direction, The variance between the two vectors is then used to describe the resultant R.

As a result, the Vector Subtraction formula is  $\bar{R}$ = $\bar{A}$ - $\bar{B}$

Note: Any of the concepts and formulae discussed in this vector formula sheet can come in useful when learning about three-dimensional geometry.

A 3D Geometry vector formula sheet is also available on every website.

### Examples of Vector Formula

Q.1) Find the Addition and Subtraction of Given Vectors.

1. (2,3,4) and (5,7,8)

2. (6,3,2) and (7,5,3)

Answer:

By using the triangular law of addition the given vectors are,

a) (2,3,4) and (5,7,8)

⇒ {2+5,3+7,4+8}

⇒ {7,10,12}

b) (6,3,2) and (7,5,3)

⇒ {6+7,3+5,2+3}

⇒ {13,8,5}

By using the vector subtraction law the given vector is,

a) (2,3,4) and (5,7,8)

⇒ {2-5,3-7,4-8}

⇒ {-3,-4,-4}

b) (6,3,2) and (7,5,3)

⇒ {6-7,3-5,2-3}

⇒ {-1,-2,-1}

## FAQs on Vector Formulas

Q.1) What is a Vector Quantity?

Answer: In physics, a quantity has both magnitude and direction. It's usually represented by an arrow of the same direction as the quantity and a length equal to the magnitude of the quantity.

Q.2) What is a Three-dimensional Vector?

Answer: A 3D vector is a three-dimensional line segment that runs from point A (tail) to point B. (head). A magnitude (or length) and direction are assigned to each vector.

Q.3) How Do You Represent a Force in Vector Form?

Answer: A force vector is a representation of both the magnitude and direction of a force. This is, in contrast, to simply stating the force's magnitude, which is referred to as a scalar quantity. A vector is typically represented by an arrow pointing in the force's direction and has a length proportional to the magnitude of the force.