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Determine the head of the vector whose tail is given. Make a sketch.

– 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 $(2,2)$

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

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

\[p=3;q=4\]

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