Equation of a Line in Space

In three-dimensional space, a line's position and orientation can be defined using vectors and coordinates. A line in 3D space is uniquely determined in one of two ways: either it passes through a specific given point and heads in a specific direction, or it passes through two given points.

To find the equation of a line passing through a given point and parallel to a given direction vector, we use two main forms:

• Vector Form: Let the line pass through a point with position vector \[\vec{a}\] and be parallel to a vector \[\vec{b}\]. If \[\vec{r}\] is the position vector of any arbitrary point on the line, the vector equation is:

  \[\vec{r} = \vec{a} + \lambda\vec{b}\]

• Cartesian Form:

  If the line passes through a point \[A(x_1, y_1, z_1)\] and has a direction vector with direction ratios a, b, c, the Cartesian equation is:

  \[\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\]

   

  

Question: Find the vector and Cartesian equations of the line passing through the point (5, 2, -4) and parallel to the vector \[3\hat{i} + 2\hat{j} - 8\hat{k}\].

Solution:

• Vector Equation: We know \[\vec{a} = 5\hat{i} + 2\hat{j} - 4\hat{k}\] and \[\vec{b} = 3\hat{i} + 2\hat{j} - 8\hat{k}\].

Plugging this into \[\vec{r} = \vec{a} + \lambda\vec{b}\], we get:

• Cartesian Equation: The point is \[(x_1, y_1, z_1) = (5, 2, -4)\] and the direction ratios are a = 3, b = 2, c = -8.

  Plugging this into the formula gives:

  \[\frac{x - 5}{3} = \frac{y - 2}{2} = \frac{z + 4}{-8}\]

• Through point \(\vec a\) and parallel to \(\vec b\): \(\vec r = \vec a + \lambda \vec b\).

• Parametric form: \(x = x_1 + \lambda a,; y = y_1 + \lambda b,; z = z_1 + \lambda c\).

• Cartesian form: \(\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}\).

• Through two points: \(\vec r = \vec a + \lambda(\vec b-\vec a)\).