**– Given vector **

**\[ \ \left[\begin{matrix}-2\\5\\\end{matrix}\right]\ \]**

**– Tail of the vector is $( -3, 2) $**

**\[ \ \left[\begin{matrix}-3\\2\\\end{matrix}\right]\ \]**

In this question, we have to find the **head of the vector** when the **vector** and **its tail** are given.

The basic concept behind this question is the knowledge of **vectors, subtraction addition, **and **multiplication **of the** vector.**

## Expert Answer

Given **vector** we have:

\[ \ \left[\begin{matrix}-2\\5\\\end{matrix}\right]\ \]

Let us suppose the head of the given matrix is:

\[ \ \left[\begin{matrix}p\\q\ \\\end{matrix}\right]\ \]

Now given in the question **statement** we have the **tail of the matrix** which is $ ( -3, 2) $ this can be **expressed** in the form of a **matrix** as:

\[ \ \left[\begin{matrix}-3\\2\\\end{matrix}\right]\ \]

As we know, the** vector matrix** is equal to the** tail of the vector-matrix** subtracted from the **head of the vector matrix**. So we can write the above notation in the **form of matrices** as below:

\[ \left[\begin{matrix}-2\\5\\\end{matrix}\right]\ =\ \left[\begin{matrix}p\\q\ \\\end{matrix}\right]\ -\ \left[\begin{matrix}-3\\2\\\end{matrix}\right]\ \]

Subtracting the **tail of the vector-matrix** from the **head of the vector matrix**, we get:

\[ \left[\begin{matrix}-2\\5\\\end{matrix}\right]\ =\ \left[\begin{matrix}p+3\\q\ -\ 2\\\end{matrix}\right] \]

Now equating the equations, put the **first equation** equal to the first element on the other side of the **equality sign**. We have the following expression:

\[ -2 = p + 3 \]

\[ p + 3 = -2 \]

Solving for the **value of $ p$,** we get:

\[ p + 3 = -2 \]

\[ p = -2 – 3 \]

\[ p = -5 \]

So we get the value of the supposed variable $ p $ in the** head vector** as $ -5$. Now to find the other variable $ q $, put the **second equation** equal to the second element of the matrix on the other side of the **equality sign**. Thus, we have the following expression:

\[ 5 = q – 2 \]

\[ q – 2 = 5 \]

Solving for the** value of $ q $,** we get:

\[ q -2 = 5 \]

\[ q = 5 + 2 \]

\[q=7\]

So we get the **value** of the supposed variable $ q $ in the **head vector** as $ 7 $.

Now our required** head of the vector** will be $( -5, 7)$ and it will be expressed in the **form of a vector** as:

\[ \ \left[\begin{matrix}p\\q\ \\\end{matrix}\right]\ = \ \left[\begin{matrix}-5\\7\ \\\end{matrix}\right]\ \]

## Numerical Result

Suppose the **head** of the given matrix is:

\[ \ \left[\begin{matrix}p\\q\ \\\end{matrix}\right]\ \]

We get the value of the **supposed variable $ q $** in the head vector as $ 7 $. which is:

\[q=7\]

And also we get the **value of the supposed variable $ p $** in the head vector as $ -5$, so:

\[p=-5\]

Now our required **head of the vector** will be $( -5, 7)$ and it will be expressed in the **form of a vector** as:

\[ \ \left[\begin{matrix}p\\q\ \\\end{matrix}\right]\ = \ \left[\begin{matrix}-5\\7\ \\\end{matrix}\right]\ \]

## Example

Find** head of the vector** $(1,2)$ whose tail is $(2,2)$

\[\left[\begin{matrix}1\\2\\\end{matrix}\right]\ =\ \left[\begin{matrix}p\\q\ \\\end{matrix}\right]\ -\ \left[\begin{matrix}2\\2\\\end{matrix}\right]\]

\[\left[ \begin{matrix}1\\2\\\end{matrix}\right]\ =\ \left[\begin{matrix}p-2\\q-2\\\end{matrix}\right]\]

\[p=3;q=4\]