# Cartesian Coordinates on a Line

## Position of a Point on a Straight Line

Cartesian coordinates for a one-dimensional space are defined as follows. Take a straight line and choose an arbitrary point *O* as the origin. Specify a unit of measure which allows us to measure the lengths of segments of the line.

To uniquely determine the *x*-coordinate of any point *M*, we also need to specify a certain direction of the line. It is usually assumed that *x* > 0 the point *M* is located to the right of the origin *O* and, accordingly, *x* < 0 if the point *M* is located to the left. We will mark the positive direction of the coordinate line with an arrow.

The fact that point *M* has coordinate *x* is denoted as \(x_M,\) \(x_1\) or \({M\left({x}\right),}\) etc. The origin *O* has coordinate \(x = 0.\)

By introducing Cartesian coordinates on the line, any point \(M\) on the straight line is associated with a well-defined real number \(x.\) There is a one-to-one correspondence between the real numbers and points on a line.

The one-dimensional coordinate system introduced in this way is also known as a number line.

## Displacement

Consider two arbitrary points *M*_{1} and *M*_{2} with coordinates *x*_{1}, *x*_{2}, respectively.

The displacement of point *M*_{2} with respect to point *M*_{1} in a given one-dimensional coordinate system is determined by the formula

## Distance Between Two Points

The distance \(d\left({M_1,M_2}\right)\) between points \(M\left({x_1}\right)\) and \(M\left({x_2}\right)\) on the coordinate line is defined by the formula

## Center of Gravity of a System of Material Points

Let points \(M_1, M_2,\ldots,M_n\) have coordinates \(x_1, x_2,\ldots,x_n\) and masses, respectively, \(m_1, m_2,\ldots,m_n.\) Then the coordinate of the center of gravity of such a system is calculated by the formula

## Solved Problems

### Example 1.

Given the coordinates of three points \(M_1\left({-3}\right),\) \(M_2\left({-7}\right),\) \(M_3\left({2}\right)\) on a line. Determine the displacement and distance for all pairs of points.

Solution.

The figure below shows how the given points are located on the straight line:

The displacement is a signed quantity. Therefore we have \(6\) possible values:

The distance is defined as the absolute value of displacement. So we only have three distance values:

### Example 2.

Indicate on the number line all points whose coordinates \(x\) satisfy the inequality

Solution.

This double inequality has the following solution:

On the coordinate axis, the solution is depicted as two segments:

The answer is written as the set

### Example 3.

Find points on the number line whose coordinates \(x\) satisfy the equation

Solution.

We replace the absolute value equation with two equivalent equations:

Solve both these equations:

Thus there are two points \(x = -6,\) \(x = 0\) that satisfy the given equation.

### Example 4.

Which of the two points lies to the right: \(M\left({x}\right)\) or \(L\left({2x}\right)?\)

Solution.

If \(x \gt 0,\) then

that is, the point \(L\) lies to the right of the point \(M.\)

If \(x \lt 0,\) then

that is, the point \(L\) lies to the left of the point \(M.\)

If \(x = 0,\) then

In this case both points coincide.

### Example 5.

Three material points with masses \(m_1 = 3, m_2 = 2, m_3 = 1\) are located on the number line and have coordinates \(x_1 = 1,\) \(x_2 = 2,\) \(x_3 = 3,\) respectively. Find the coordinate of the center of gravity of the system.

Solution.

Calculate the coordinate \(x\) of the center of gravity using the formula

Substitute known values:

### Example 6.

Indicate on the number line the set of points whose coordinates satisfy the inequality

Solution.

We first solve the quadratic equation and plot the graph of the function.

To get rid of the absolute value consider two cases.

#### Case 1.

If \(x \in \left({-\infty, 1}\right] \cup \left[{4, \infty}\right),\) then \(x^2 - 5x + 4 \ge 0\) and the inequality takes the form

Obviously this inequality has no solutions: \(x \in \varnothing.\)

#### Case 2.

If \(x \in \left({1, 4}\right),\) then \(x^2 - 5x + 4 \lt 0\) and the inequality is written as

Answer: \(x \in \left({1, 4}\right).\)