Equation of a line
Equation of a line on plane
General form of a line equation
Any equation of a line on plane can by written in the general form
A x + B y + C = 0
where A and B are not both equal to zero.
Slope intercept form of a line equation
The general equation of a line when B ≠ 0 can be reduced to the next form
y = k x + b
where k is the slope of the line and b is the y-intercept. Slope of the line is equal to the tangent of the angle between this line and the positive direction of the x-axis. The y-coordinate is the location where line crosses the y-axis.
k = tg φ
Equation of the line passing through two different points on plane
If the line passes through two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}), such that x_{1} ≠ x_{2} and y_{1} ≠ y_{2}, then equation of line can be found using the following formula
x - x_{1} | = | y - y_{1} |
x_{2} - x_{1} | y_{2} - y_{1} |
Parametric equations of a line on plane
Parametric equation of the line can be written as
x = l t + x_{0} | |
y = m t + y_{0} |
where N(x_{0}, y_{0}) is coordinates of a point that lying on a line, a = {l, m} is coordinates of the direction vector of line.
Canonical equation of a line on plane
If you know the coordinates of the point A(x_{0}, y_{0}) If you know the coordinates of the point n = {l; m}, then the equation of the line can be written in the canonical form using the following formula
x - x_{0} | = | y - y_{0} |
l | m |
Example 1. Find the equation of a line passing through two points A(1, 7) and B(2, 3).
Solution. We use the formula for the equation of a straight line passing through two points
From this equation, we express y in terms of x
Find the Slope intercept form of line equation.
Multiply equations by -4.
y - 7 = -4(x - 1)
y = -4x + 11
Find the Parametric equations of this line
We use MN as direction vector of line.
MN = {2 - 1; 3 - 7} = {1; -4}
We use coordinates of point М in parametric equations of line
Solution. It is impossible to use Equation of the line passing through two different points, since M_{y} - N_{y} = 0.
Find the Parametric equations of this line. We use MN as direction vector of line.
MN = {2 - 1; 3 - 3} = {1; 0}
We use coordinates of point М in parametric equations of line
Equation of a line in space
Equation of the line passing through two different points in space
If the line passes through two points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}), such that x_{1} ≠ x_{2}, y_{1} ≠ y_{2} and z_{1} ≠ z_{2}, then equation of line can be found using the following formula
x - x_{1} | = | y - y_{1} | = | z - z_{1} |
x_{2} - x_{1} | y_{2} - y_{1} | z_{2} - z_{1} |
Parametric equations of a line in space
Parametric equation of the line can be written as
x = l t + x_{0} | |
y = m t + y_{0} | |
z = n t + z_{0} |
where (x_{0}, y_{0}, z_{0}) is coordinates of a point that lying on a line, {l; m; n} is coordinates of the direction vector of line.
Canonical equation of a line in space
If you know the coordinates of the point A(x_{0}, y_{0}, z_{0}) that lies on the line and the direction vector of the line n = {l; m; n}, then the equation of the line can be written in the canonical form using the following formula.
x - x_{0} | = | y - y_{0} | = | z - z_{0} |
l | m | n |
Straight line as crossings of two planes
If the line is the intersection of two planes, then the equation of line can be found as the solution of the following system of equations
A_{1}x + B_{1}y + C_{1}z + D_{1} = 0 | |
A_{2}x + B_{2}y + C_{2}z + D_{2} = 0 |
on condition that there has not been equality
A_{1} | = | B_{1} | = | C_{1} | . |
A_{2} | B_{2} | C_{2} |
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