Equation of a Line in Space

Assume that we know a line that runs from south-west to north-east. In other words, we know the line’s orientation. Is this criteria adequate to identify the line’s uniqueness?

No. See how many parallel lines may exist, all flowing from south-west to north-east. As a result, the line is not determined just by its direction.

However, we now know that the line goes through a certain point P. Can we now determine the line in a unique way?

Yes. You’ll see that there’s only one line that runs from south to north and also crosses through P. So, If we know the direction of a line and the point through which it travels, we can determine it uniquely.

Straight Line’s Parametric Equation in Space

A line in space’s parametric equations are a non unique set of three equations of the type

x = xa + tl.

y = ya + tm.

z = za + tn.

where (x,y,z) are the coordinates of a point on the line, (l,m,n) is the line’s direction vector, and t is a real value (the parameter) ranging from -∞ to +∞.

Slope of a line

The change in y coordinate with respect to the change in x coordinate is known as the slope of a line in mathematics.

Δy represents the net change in y-coordinate, while Δx represents the net change in x-coordinate.

As a result, the y-coordinate change with regard to the x-coordinate change is given by,

m = change in y/ change in x = Δy/Δx.

It can also be represented as 

tanθ = Δy/Δx

Where m and tanθ represent the slope

Vector Equation

Case 1: Line is parallel to a vector and passes through a given point

Consider a line that passes through a point with the position vector a^→  and is parallel to the vector d^→. Notice how the vector d^→ defines the line’s ‘direction.’ Let r^→ be the position vector of a general point on the line. Then the equation of the line can be written as

r^→ = a^→ + λd^→

can range from -∞ to +∞, thus covering all the points on the line

Case 2: Line Passes through 2 given points 

Consider a line which passes through 2 points with the position vectors a^→ and b^→ , the line will be parallel to b^→–a^→, so we can replace d in the previous expression with b^→–a^→ to get the equation of the line

r^→ = a^→ + λ(b^→–a^→)

Cartesian Equation

Case 1: Line is parallel to a vector and passes through a given point

Consider a line that crosses a point P(x1, y1, z1) and is parallel to the vector ai^+bj^ +ck^, Then the direction ratio is a:b:c

Then every point on the line can be expressed in term of P as 

x = x1 + λa

y = y1 + λb

z = z1 + λc

Upon eliminating λ, we get the equation of the line as

(x-x1)/a = (y-y1)/b = (z-z1)/c

Case 2: Line Passes through 2 given points 

Consider a line that passes through 2 points P (x1, y1, z1) and Q (x2, y2, z2). Then the direction ratio will be (x2-x1):(y2-y1):(z2-z1)

So the Equation of the line can be expressed as 

(x-x1)/(x2-x1) = (y-y1)/(y2-y1) = (z-z1)/(z2-z1)

Conclusion

A straight line is defined as the set of all points connecting and extending beyond two points. Furthermore, straight lines in Euclidean geometry have only one dimension, which is length, and they stretch in two directions indefinitely. The general equation for calculating a straight line is y = mx + c, where m is the gradient and y = c is the value at which the line crosses the y-axis. The intercept on the y-axis is also known as the value of c or the number c. A straight line with gradient m and intercept c on the y-axis has the equation y = mx + c.