BASIC MATHEMATICS FOR NATURAL SCIENCES UNDERGRADUATE STUDENT TEXTBOOK

If P and Q are two points on the coordinate plane, then PQ represents the line segment joining P and Q and d(P,Q) or |PQ| represents the distance between P and Q.

Recall that the distance between points a and b on a number line is |a -b| = |b −a|. Thus, the distance between two points P(\(x_1\text{,}\) \(y_1\)) and R(\(x_2\text{,}\) \(y_1\)) on a horizontal line must be |\(x_2\) \(x_1\)| and the distance between Q(\(x_2\text{,}\) \(y_2\)) and R(\(x_2\text{,}\) \(y_1\)) on a vertical line must be |\(y_2\) -\(y_1\)|.(See, Figure 4.1.1) .

To find distance \(|PQ|\) between any two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\text{,}\) we note that triangle PRQ in Figure 4.1.1 is a right triangle, and so by Pythagorean Theorem we get:

\(|PQ|^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \iff |PQ| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Therefore, we have the following:

\(\textbf{Distance Formula:}\) The distance between the points P(\(x_1\text{,}\) \(y_1\)) and Q(\(x_2\text{,}\) \(y_2\)) is

Note that, from the distance formula, the distance between the origin O(0,0) and a point P(x, y) is

Division point of a line segment: Given two distinct points P(\(x_1\text{,}\) \(y_1\)) and Q(\(x_2\text{,}\) \(y_2\)) in the coordinate plane, we want to find the coordinates (\(x_0\text{,}\) \(y_0\)) of the point R that lies on the segment PQ and divides the segment in the ratio \(r_1\) to \(r_2\) ; that is

where \(r_1\) and \(r_2\) are given positive numbers. (Figure 4.1.3)

To determine (\(x_0\text{,}\) \(y_0\)), we construct two right triangles ∆PSR and ∆RTQ as shown. We then have |PS| = \(x\) −\(x\) , |SR| = \(y_0\)−\(y_1\) , |RT| = x_2 −x_0 , and |TQ| = y_2 −y_0 . Now since ∆PSR is similar to ∆RTQ, we have that

or \(r_2\)(\(x_0\) −\(x_1\)) = \(r_1\) (\(x_2\) −\(x_0\)) and \(r_2\)(\(y\) −\(y\)) = \(r_1\) (\(y_2\) −\(y_0\)) .

Solving for \(x_0\) and \(y_0\) , we obtain

Therefore, we have shown the following.

An equation of a line l is an equation which must be satisfied by the coordinates (x, y) of every point on the line. A line can be vertical, horizontal or oblique. The equation of a vertical line that intersects the x-axis at (a, 0) is x=a because the x-coordinate of every point on the line is a. Similarly, the equation of a horizontal line that intersects the y-axis at (0, b) is y=b because the y-coordinate of every point on the line is b.

An oblique line is a straight line which is neither vertical nor horizontal. To find equation of an oblique line we use its slope which is the measure of the steepness of the line. In particular, the slope of a line is defined as follows.

Thus the slope of a line l is the ratio of the change in y, ∆y, to the change in x, ∆x (See Figure 4.1.11). Hence, slope is the rate of change of y with respect x. The slope depends also on the angle of inclination of the line. Note that the angle of inclination θ is the angle between x-axis and the line (measured counterclockwise from the direction of positive x-axis to the line). Observe that

Therefore, if θ is the angle of inclination of a line, then its slope is m = tanθ.

Now let us find an equation of the line that passes through a point \(P_1\)(\(x_1\text{,}\) \(y_1\)) and has slope m. A point P(x, y) with x≠\(x_1\) lies on this line if and only if the slope of the line though \(P_1\) and P is m; that is

This leads to the following equation of the line:

In general, depending on the given information, you can show that the equations of oblique lines can be obtained using the following formulas.

\(\textbf{General Form:}\) In general, the equation of a straight line can be written as

for constants a, b, c with a and b not both zero. Indeed, if a=0 the line is a horizontal line given by y = −c/b, if b=0 the line is a vertical line given by x = −c/a, and if both a, b≠ 0 it is the oblique line given by \(y=-\left ( \frac{a}{b} \right )x-\frac{c}{b}\) with slope \(m=-\frac{a}{b}\) and y-intercept \(y=-\frac{c}{b}\) .

\(\textbf{Parallel and Perpendicular lines:}\) slopes can be used to check whether lines are parallel, perpendicular or not. In particular, let \(I_1\) and \(l_2\) be non-vertical lines with slope \(m_1\) and \(m_2\text{,}\) respectively. Then,

Moreover, if \(l_1\) and \(l_2\) are are both vertical lines then they are parallel. However, if one of them is horizontal and the other is vertical, then they are perpendicular.

Suppose a line \(l\) and a point P(x,y) not on the line are given. The distance from P to \(l\text{,}\) d(P, \(l\)), is defined as the perpendicular distance between P and l . That is,

d(P, \(l\)) = |PQ|, where Q is the point on \(l\) such that PQ⊥ \(l\text{.}\)(See Figure 4.1.19)

If P is on \(l\text{,}\) then d(P, \(l\)) = 0. Moreover, given a point P(h,k) observe that

In general, however, to find the distance between a point P(\(x_0\)\(, y_0\)) and an arbitrary line \(l\) given by ax + by +c = 0, we have to first get a point Q on \(l\) such that PQ⊥ \(l\) and then compute |PQ|. This yields the formula given in the following Theorem.

In particular, if we take (\(x_0\)\(,y_0\))=(0,0) in this formula, we obtain the distance between the origin O(0,0) and a line L : ax + by +c = 0 which is given by