Representation of Vectors

 

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.

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 $\overrightarrow{AB}$.

The above vector can also be represented by - $\overrightarrow{BA}$ which is the negative of the vector $\overrightarrow{AB}$.

Sometimes vectors are denoted by a single small letter also with an arrow over it. For example, we can label the above vector $\overrightarrow{AB}$ 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 $\overrightarrow{AB}$ 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.
• $\hat{a}$ 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.