Gradient Of a Line

The gradient of a line is defined as the change in the "y" coordinate with respect to the change in the "x" coordinate of that line. The gradient of a line is helpful to find the inclination or steepness of a line. The gradient of a line formula calculates the slope of any line by finding the ratio of the change in the y-axis to the change in the x-axis.

Let us learn more about the gradient of a line, how to find the gradient of a line, applications of gradient of a line, and the examples.

The gradient of a line is used to calculate the steepness of a line and determines how much it is inclined with reference to the x-axis. To calculate the gradient of any line, the x and y coordinates of a line are used. In other words, it is the ratio of the change in the y-axis to the change in the x-axis.

The formula to calculate the gradient of a line is given as, m = (\(y_2\)−\(y_1\))/(\(x_2\)−\(x_1\)) = Δy/Δx, Where m represents the gradient of the line. \(x_1\), \(x_2\) are the coordinates of the x-axis, and \(y_1\), \(y_2\) are the coordinates of the y-axis. The x and y coordinates of the point are used to calculate the gradient of the line, which is also referred as the slope of the line.

The gradient of a line can also be measured as the net change in y coordinate with respect to the change in x coordinate and can be written as, m = Δy/Δx. Here, Δy is the change in y-coordinates, and Δx is the change in the x-coordinates. Also, we know that tan θ is also the slope of the line where θ is the angle made by the line with the positive direction of the x-axis, and, tanθ=height/base.Thus, the gradient of the line is, m=tanθ=Δy/Δx

The gradient of a line can be computed from the following below methods.

• From Two Points: The gradient of a line can be computed from any two points lying on the line. For the two points \((x_1, y_1)\), \((x_2, y_2)\), the gradient of the line is \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)
• Angle Of Inclination: The angle of inclination of the line is θ with respect to the x-axis. The gradient of a line is the value obtained from the tangent of the angle of inclination of the line with respect to the x-axis. m = Tanθ.
• Equation of a Line: The equation of a line is also helpful to find the gradient of a line. The standard form of the equation of a line, having the equation ax + by + c = 0, has a gradient of m = -a/b. And the slope-intercept form of the equation of a line y = mx + c, has a gradient m, which is the coefficient of the x term.

The gradient of a line has numerous applications in coordinate geometry, three-dimensional geometry, and vector algebra. Some of the important applications of the gradient of a line are as follows.

• The gradient of a line gives the inclination of the line with respect to the x-axis.
• The gradient of a line is used to find the equation of a line.
• The gradient of a line has applications in three-dimensional geometry, to find the equation of a line and the equation of a plane.
• The gradient of two lines is useful to find the angle between the two lines.
• The gradient of two lines is useful to know if the two lines are parallel or perpendicular with respect to each other.
• The product of the gradient of two perpendicular lines is equal to -1. \(m_1.m_2 = -1\).
• The gradient of two parallel lines is equal in value. \(m_1 = m_2\).

Related Topics

The following topics help in a better understanding of the gradient of a line.

• Coordinate Geometry
• Slope
• Slope Formula
• Point Slope Form
• Slope Intercept Form
• Angle Between Two Lines

What Is Gradient of a Line?

The gradient of a line is the slope or inclination of the line with respect to the x-axis. The gradient of a line is given as, m = (\(y_2\)−\(y_1\))/(\(x_2\)−\(x_1\)) = Δy/Δx, Where m represents the gradient of the line. \((x_1, y_1)\). \((x_2, y_2)\) are any two points on the line..

How Do You Find Gradient of a Line?

The gradient of a line can be computed from any two points on the line, the angle of inclination of the line, or from the equations of the line. .

• For the two points \((x_1, y_1)\), \((x_2, y_2)\), the gradient of the line is \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)
• For the angle of inclination of the line 'θ' with respect to the x-axis, the gradient of a line is m = Tanθ.
• The standard equation of a line ax + by + c = 0, has the gradient of m = -a/b.
• The slope-intercept form of the equation of a line y = mx + c, has a gradient m, which is the coefficient of the x term.

How Do You Find Gradient of a Line From Two Given Points?

The gradient of a line from the two given points, is equal to the ratio of the change of the y coordinates, to the change of the x coordinates. For any two points \((x_1, y_1)\), \((x_2, y_2)\), the gradient of the line is \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\).

What Is The Formula For Gradient of a Line?

The formula to find the gradient of a line is \(m = \dfrac{y_2 - y_1}{x_2 - x_1} = Tanθ\).