## 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.

A vector is an object that has both a magnitude and a direction. In Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction.

Vector is typically represented by an arrow whose direction is the same as that of the quantity and whose length is proportional to the quantity magnitude. Vector quantities are often represented by scaled vector diagrams. The vector diagram depicts a displacement vector.

## 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

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:

### 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.

### 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.

### 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.

### 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.

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.

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

(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.

**Triangular law of addition**

If two forces Vector A and Vector B are acting in the same direction, then its resultant R will be the sum of two vectors.

**Parallelogram law of addition **

If two forces Vector A and Vector B are represented by the adjacent sides of the parallelogram, then their resultant is represented by the diagonal of a parallelogram drawn from the same point.

**Solved examples of vector**

**Example: **Give the vector for each of the following:

(a) The vector from (2, -7, 0) to (1, -3, -5)

(b) The vector from (1, -3, -5) to (2, -7, 0)

(c) The position vector for (-90, 4)

**Solution: **

(a): Remember that to construct this vector we subtract coordinates of the starting point from the ending point.

{1 -2, -3 – (-7), -5 -0} = {-1, 4, -5}

(b): Same thing here: {2 -1, -7 – (-3), 0 (-5)} = {1, -4, 5}

Notice that the only difference between the first two is the signs are all opposite. This difference is important as it is this difference that tells us that the two vectors point in opposite directions.

(c): Not much to this one other than acknowledging that the position vector of a point is nothing more than a vector with the point’s coordinates as its components. (-90, 4)

## What are the List of Vector Formulas?

The list of vector formulas includes formulas performing the operations for a single vector and across the vectors. The formulas of direction ratios, direction cosines, the magnitude of a vector, unit vector are performed on the same vector. And the formulas of dot product, cross product, projection of vectors, are performed across two vectors.

## Solved Examples on Vector Formulas

**Solved Examples on Vector Formula**

Q.1: Provide the vector for each of the following:

(a) The vector from the position (2, -7, 0) to position (1, -3, -5)

(b) The vector from position (1, -3, -5) to position (2, -7, 0)

(c) The position vector for (-90, 4)

Solution:

(a): As we know that to construct this vector we subtract coordinates of the starting point from those of the ending point.

{ 1 -2, -3 – (-7), -5 -0 } = {-1, 4, -5}

(b): Similarly, we may do here:

{ 2 -1, 7 – (-3), 0 (-5) } = { 1, -4, 5 }

Remember that the only difference between the first two is the signs, which are all opposite. This difference is important as it is saying about the two vectors which point in opposite directions.

(c): Nothing is needed to do here much to this one other than acknowledging that the position vector of a point is nothing more than a vector with the point’s coordinates as its components. Thus it is ( -90, 4 )

## List of Formulas for Vectors

Note: More than two vector also can be added similar to the above method.

**Mathematical method:**

cos α, cos β and cos γ are called direction cosines in x direction, y direction and z direction respectively

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