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Representation of Vector
The representation of vector is done by using the arrow. We know that an arrow contains a head and a tail. The head of the arrow denotes the direction of the vector. Let us learn more about the representation of vector along with examples.
1.  What is Representation of Vector? 
2.  Representation of Position Vectors 
3.  Representation of Vector Magnitude 
4.  Representation of Vector Operations 
5.  FAQs on Representation of Vector 
What is Representation of Vector?
The representation of vector is done by a directed line segment. It is an arrow that has a head and a tail. here,
 The starting point of the vector is called its tail (or) the initial point of the vector.
 The ending point of the vector is called its head (or) the terminal point of the vector.
The head of the vector shows its direction. The direction of the vector is the angle made by it with a reference line. A vector that starts from a point A and ends at a point B is denoted by \(\overrightarrow{A B}\).
The above vector can also be represented by  \(\overrightarrow{B A}\) which is the negative of the vector \(\overrightarrow{A B}\).
Sometimes vectors are denoted by a single small letter also with an arrow over it. For example, we can label the above vector \(\overrightarrow{A B}\) as \(\overrightarrow{a}\). But we cannot identify the initial point and the final point of the vector in this representation though.
Representation of Position Vectors
The points on a coordinate plane is represented by position vectors. In the above figure,
 The point A is represented by the position vector \(\overrightarrow{OA}\) = <3, 2>
 The point B is represented by the position vector \(\overrightarrow{OB}\) = <2, 1>
Here, O is the origin. We can calculate the components of a vector \(\overrightarrow{A B}\) by subtracting the position vector of initial point from that of the terminal point. In the above figure,
\(\overrightarrow{AB}\) = \(\overrightarrow{OB}\)  \(\overrightarrow{OA}\)
= <2, 1>  <3, 2>
= <2 (3), 1  2>
= <5, 1>
Thus, the vector that represents \(\overrightarrow{AB}\) in the above figure is <5, 1>.
Representation of Vector Magnitude
The magnitude of a vector \(\overrightarrow{AB}\) is represented by either \(\overrightarrow{AB}\) or simply AB. We use AB to represent the magnitude of the vector \(\overrightarrow{AB}\) because the magnitude is nothing but its length and AB represents the length of the line segment connecting A and B. In the same way, the magnitude of a vector \(\overrightarrow{a}\) is represented as either \(\overrightarrow{a}\) or simply 'a'. We can find the magnitude of a vector (when we know its components) by taking the square root of the sum of the squares of the components. From the last example,
\(\overrightarrow{AB}\) = <5, 1>
Its magnitude is,
\(\overrightarrow{AB}\) (or) AB =√(5)² + (1)² = √26
Representation of Vector Operations
For any three vectors \(\overrightarrow{a}\), \(\overrightarrow{b}\), and \(\overrightarrow{c}\):
 \(\overrightarrow{a}\) + \(\overrightarrow{b}\) represents the sum of vectors.
 \(\overrightarrow{a}\)  \(\overrightarrow{b}\) represents the difference of vectors.
 \(\overrightarrow{a}\) · \(\overrightarrow{b}\) represents the dot product of vectors.
 \(\overrightarrow{a}\) × \(\overrightarrow{b}\) represents the cross product of vectors.
 \(\widehat{a}\) represents the unit vector in the direction of \(\overrightarrow{a}\).
 [\(\overrightarrow{a}\) \(\overrightarrow{b}\) \(\overrightarrow{c}\)] (or) \(\overrightarrow{a}\) · [ \(\overrightarrow{b}\) × \(\overrightarrow{c}\) ] (or) [\(\overrightarrow{a}\) × \(\overrightarrow{b}\) ] · \(\overrightarrow{c}\) is called the scalar triple product.
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Representation of Vector Examples

Example 1: If a vector starts at a point P = <5, 4, 3> and ends at <3, 4, 0>, then how do we represent it?
Solution:
We know that a vector is represented by an arrow which is placed over starting point that is immediately followed by the ending point of the vector.
Since the starting and ending points of the given vector are P and Q respectively, the vector is \(\overrightarrow{PQ}\).
Answer: \(\overrightarrow{PQ}\).

Example 2: Find the components of vector \(\overrightarrow{PQ}\) that is mentioned in Example 1.
Solution:
The given position vectors are:
\(\overrightarrow{OP}\) = <5, 4, 3>
\(\overrightarrow{OQ}\) = <3, 4, 0>
We know that:
\(\overrightarrow{PQ}\) = \(\overrightarrow{OQ}\)  \(\overrightarrow{OP}\)
= <3, 4, 0>  <5, 4, 3>
= <3  (5), 4  4, 0  3>
= <8, 8, 3>Answer: <8, 8, 3>.

Example 3: Find the magnitude of \(\overrightarrow{PQ}\) and also show how can it be represented.
Solution:
The magnitude of \(\overrightarrow{PQ}\) is represented as \(\overrightarrow{PQ}\) (or) PQ.
PQ = √8² + (8)² + (3)²
= √64+64+9
= √137Answer: PQ (or) \(\overrightarrow{PQ}\) = √137
FAQs on Representation of Vector
How to Represent a Vector on a Diagram?
The representation of a vector is done by using an arrow whose tail is the initial point of the vector and whose head is the terminal point of the vector. A vector that starts at P and ends at Q is denoted by \(\overrightarrow{PQ}\).
How to Represent a Vector on a Graph?
A vector has initial and final points. To represent it on the graph, just plot its initial and final points on the graph sheet and join them by an arrow such that arrow's tail is at the initial point and its head is at the final point.
How to Represent a Vector magnitude?
A vector's magnitude is represented by using the symbol of "modulus". i.e., the magnitude of a vector \(\overrightarrow{AB}\) is denoted by \(\overrightarrow{AB}\) . We can represent it simply without modulus symbol and without vector sign (arrow), i.e., \(\overrightarrow{AB}\) = AB.
How to Represent a Vector Product?
Vector product is also known as the cross product and the cross product of two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is denoted by \(\overrightarrow{a}\) × \(\overrightarrow{b}\).
What does Vector Sum Represent?
By triangle law of addition, the sum of two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is the vector that starts from the tail of \(\overrightarrow{a}\) and ends at the head of \(\overrightarrow{b}\).
What Does Vector Difference Represent?
The difference of two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is the vector that starts at the head of \(\overrightarrow{b}\) and ends at the head of \(\overrightarrow{a}\).
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