# Equation of a Line in 3D

We all know the very popular equation of the straight line* Y = m . X + C *which a straight line in a plane. But here we are going to discuss the Equation of a Straight Line in 3-dimensional space. A

**Straight**Line is uniquely characterized if it passes through the two unique points or it passes through a unique point in a definite direction. In Three Dimensional Geometry lines (straight lines) are usually represented in the two forms Cartesian Form and Vector form. Here we are going to discuss the two-point form

**of a straight line in 3-dimensions using both cartesian as well as vector form.**

**Equation of a Straight Line in Cartesian Form**

For writing the equation of a straight line in the cartesian form we require the coordinates of a minimum of two points through which the straight line passes. Let’s say * (x_{1}, y_{1}, z_{1})* and

*are the position coordinates of the two fixed points in the 3-dimensional space through which the line passes.*

**(x**_{2}, y_{2}, z_{2})**Now to obtain the equation we have to follow these three steps:**

**Step 1:**Find the DR’s (Direction Ratios) by taking the difference of the corresponding position coordinates of the two given points.*l*_{2}– x_{1}),*m*= (y_{2}– y_{1}),*n*= (z_{2}– z_{1}); Hereare the DR’s.**l, m, n****Step 2:**Choose either of the two given points say, we choose_{1}, y_{1}, z_{1}).**Step 3:**Write the required equation of the straight line passing through the points_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}). L : (x – x_{1})/l = (y – y_{1})/m = (z – z_{1})/n

Where * (x, y, z)* are the position coordinates of any variable point lying on the straight line.

**Example 1: If a straight line is passing through the two fixed points in the 3-dimensional whose position coordinates are P (2, 3, 5) and Q (4, 6, 12) then its cartesian equation using the two-point form is given by**

**Solution:**

l = (4 – 2), m = (6 – 3), n = (12 – 5)

l = 2, m = 3, n = 7

Choosing the point P (2, 3, 5)

The required equation of the line

L : (x – 2) / 2 = (y – 3) / 3 = (z – 5) / 7

**Example 2: If a straight line is passing through the two fixed points in the 3-dimensional whose position coordinates are A (2, -1, 3) and B (4, 2, 1) then its cartesian equation using the two-point form is given by**

**Solution:**

l = (4 – 2), m = (2 – (-1)), n = (1 – 3)

l = 2, m = 3, n = -2

Choosing the point A (2, -1, 3)

The required equation of the line

L : (x – 2) / 2 = (y + 1) / 3 = (z – 3) / -2 or

L : (x – 2) / 2 = (y + 1) / 3 = (3 – z) / 2

**Example 3:** **If a straight line is passing through the two fixed points in the 3-dimensional whose position coordinates are X (2, 3, 4) and Y (5, 3, 10) then its cartesian equation using the two-point form is given by**

**Solution:**

l = (5 – 2), m = (3 – 3), n = (10 – 4)

l = 3, m = 0, n = 6

Choosing the point X (2, 3, 4)

The required equation of the line

L : (x – 2) / 3 = (y – 3) / 0 = (z – 4) / 6 or

L : (x – 2) / 1 = (y – 3) / 0 = (z – 4) / 2

**Equation of a Straight Line in** **Vector Form**

For writing the equation of a straight line in the vector form we require the position vectors of a minimum of two points through which the straight line passes. Let’s say ** **and are the position vectors of the two fixed points in the 3-dimensional space through which the line passes.

**Now to obtain the equation we have to follow these three steps:**

**Step 1:**Find a vector parallel to the straight line by subtracting the corresponding position vectors of the two given points. = (); Here is the vector parallel to the straight line.**Step 2:**Choose the position vector of either of the two given points say we choose**Step 3:**Write the required equation of the straight line passing through the points whose position vectors are**or**= + t . ()

Where ** **is the position vector of any variable point lying on the straight line and** t** is the parameter whose value is used to locate any point on the line uniquely.

**Example 1:** **If a straight line is passing through the two fixed points in the 3-dimensional whose position vectors are (2 i + 3 j + 5 k) and (4 i + 6 j + 12 k) then its Vector equation using the two-point form is given by**

**Solution:**

= (4

i+ 6j+ 12k) – (2i+ 3j+ 5k)= (2

i+ 3j+ 7k) ; Here is a vector parallel to the straight lineChoosing the position vector (2

i+ 3j+ 5k)The required equation of the straight line

L : = (2

i+ 3j+ 5k) +t. (2i+ 3j+ 7k)

**Example 2: If a straight line is passing through the two fixed points in the 3-dimensional space whose position coordinates are (3, 4, -7) and (1, -1, 6) then its vector equation using the two-point form is given by**

**Solution:**

Position vectors of the given points will be (3 i + 4 j – 7 k) and (i – j + 6 k)

= (3 i + 4 j – 7 k) – (i – j + 6 k)

= (2 i + 5 j – 13 k) ; Here is a vector parallel to the straight line

Choosing the position vector (i – j + 6 k)

The required equation of the straight line

L : = (i – j + 6 k) +

t. (2 i + 5 j – 13 k)

**Example 3:** **If a straight line is passing through the two fixed points in the 3-dimensional whose position vectors are (5 i + 3 j + 7 k) and (2 i + j – 3 k) then its Vector equation using the two-point form is given by**

**Solution:**

= (5 i + 3 j + 7 k) – (2 i + j – 3 k)

= (3 i + 2 j + 10 k) ; Here is a vector parallel to the straight line

Choosing the position vector (2 i + j – 3 k)

The required equation of the straight line

L : = (2 i + j – 3 k) +

t. (2 i + 3 j + 7 k)