Equations of a line and a plane in different forms

Terms used in the concept of the Equation of a line or a plane

• Slope – Slope is the measure of steepness or gradient of a line or a plane. It is represented by “m,” and its formula is m=y1-y2x1-x2 or tan, where is the angle between the line and the x-axis. 

• Positive/negative/zero slope – Positive slope suggests that x and y are directly proportional. A negative slope shows that x and y are inversely proportional. Zero slopes indicate that the line is parallel to the x-axis.

• Line-It is a collection of points on the Cartesian plane.

• Plane- It is the collection of lines in a 3- dimensional space.

• Tangent- It is the ratio of the sine and cosine function of a line.

• Intercept- It is the point where a line or a plane is cut or obstructed in space.

• X-intercept/y-intercept/z-intercept – x-intercept is the value of the x- coordinate where the line cuts the x-axis, and the same goes for y and z.

Equation of a line

The Equation of a line is the algebraic representation of a set of points in a two-dimensional coordinate plane. The collection of points forming a line is represented by two variables (Usually x and y), creating an equation of degree one. With the help of this Equation, we can determine whether a given point lies on the offered line. Generally, the Equation of a line is written as,

y=mx+c

Different forms of Equation of a Line

• The standard form of an equation- The standard form of an equation of a line isax+by+c=0. a and b are the coefficients, x and y are the variables, and c is constant.

• point-slope form- the point-slope form of the equation of a line is, (y – y1) = m(x – x1), where m is the slope of the given line. (x1,y1) is a point that lies on the line.

• Two-point form- the two-point form of the Equation of a line uses two points lying on the line (x1,y1) and (x2,y2). Its formula is,

 (y-y1)=(y2-y1)(x2-x1)(x-x1)

• The slope-intercept form- is the most used in this concept, as it uses only the slope m and the x-intercept c. the formula is y= mx +c

• Intercept form- is the Equation of line formed with the help of both the x-intercept and the y-intercept. Its formula is xa+yb=1, where a is the x-intercept and b is the y-intercept.

• Normal form- the normal form of the Equation or the parametric form uses parameters like a line’s sine and cosine functions. Its formula is xcosθ + ysinθ = P

• Equation of a horizontal and vertical line – the Equation of a horizontal line or a line parallel to the x-axis is x=0, and the Equation of a vertical line or a line parallel to the y- axis is y=0.

Equation of a plane

The Equation of a plane is the algebraic representation of a plane surface in a three-dimensional space. It is represented by three variables (usually x,y, and z), creating an equation of degree one in either Cartesian or Vector form. The most popular form of an equation of a plane is,

ax+by+cz=d

where at least one of the coefficients must be non-zero.

Different forms of Equation of a plane

• Perpendicular form- is the Equation of a plane with perpendicular distance d from the origin and having a normal vector n. Its formula is r . n = d

• One-point form- Is the Equation of a plane that is perpendicular to a given vector n and passing through a point a. its formula is (r-a).n=0

• Three-point form- is the equation of a plane that passes through three non-collinear points a, b and c. its formula is (r-a)[(b-a)(c-a)]=0.

• Intersection form- is the form of equation of a plane that passes through the intersection point of two different planes.If r . n1 = d1 and r . n2 = d2 are two planes , then the formula is r(n1+n2)=d1+d2

Conclusion

Important points to remember in this concept are

1. equation of the x-axis is always = 0 and equation of the y-axis is always x = 0.
2. equation of any line parallel to the x-axis is always y = b, and it intercepts the y-axis at the point (0, b).
3. equation of any line parallel to the y-axis is always x = a, and it intercepts the x-axis at the point (a, 0).
4. equation of any line parallel to ax + by + c = 0 is ax + by + d = 0.

The equation of a line perpendicular to ax + by + c = 0 is bx – ay + d = 0.