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Equation of a Line in 3D

  • Difficulty Level : Medium
  • Last Updated : 11 Feb, 2021

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 (x1, y1, z1) and (x2, y2, z2) are the position coordinates of the two fixed points in the 3-dimensional space through which the line passes.

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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 = (x2 – x1), m = (y2 – y1), n = (z2 – z1); Here l, m, n are the DR’s.
  • Step 2: Choose either of the two given points say, we choose (x1, y1, z1).
  • Step 3: Write the required equation of the straight line passing through the points (x1, y1, z1) and (x2, y2, z2). L : (x – x1)/l = (y – y1)/m = (z – z1)/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 {\vec {a}} and {\vec {n}} 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. {\vec {p}} = ({\vec {b}}-{\vec {a}} ); Here {\vec {p}} is the vector parallel to the straight line.
  • Step 2: Choose the position vector of either of the two given points say we choose {\vec {a}}.
  • Step 3: Write the required equation of the straight line passing through the points whose position vectors are {\vec {a}} and {\vec {b}}. L : {\vec {r}} = {\vec {a}} + t . {\vec {p}}  or  {\vec {r}} = {\vec {a}} + t . ({\vec {b}}-{\vec {a}})

Where {\vec {r}} 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:

{\vec {p}} = (4 i + 6 j + 12 k) – (2 i + 3 j + 5 k)

{\vec {p}} = (2 i + 3 j + 7 k) ; Here {\vec {p}} is a vector parallel to the straight line

Choosing the position vector (2 i + 3 j + 5 k)

The required equation of the straight line

L : {\vec {r}} = (2 i + 3 j + 5 k) + t . (2 i + 3 j + 7 k)

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)

{\vec {p}} = (3 i + 4 j – 7 k) – (i – j + 6 k)



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

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

The required equation of the straight line

L : {\vec {r}} = (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:

{\vec {p}} = (5 i + 3 j + 7 k) – (2 i + j – 3 k)

{\vec {p}} = (3 i + 2 j + 10 k) ; Here {\vec {p}} 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 : {\vec {r}} = (2 i + j – 3 k) + t . (2 i + 3 j + 7 k)




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