Question

Find sets of (a) parametric equations and (b) symmetric equations of the line through the two points. (For each line, write the direction numbers as integers.) $$ (2,0,2),(1,4,-3) $$

Short answer

The parametric equations of the line are \( x = 2 -t \), \( y = 4t \), \( z = 2 - 5t \) and the symmetric equations of the line are \( \frac{x - 2}{-1} = \frac{y}{4} = \frac{z - 2}{-5} \)

Step-by-step solution

Find the Direction Vector

The direction vector of the line through points P (2,0,2) and Q (1,4,-3) is pq = q - p, hence the direction vector \( \overrightarrow{pq} \) components would be the \( x, y, z \) coordinates of Q subtract the coordinates of P respectively. So, \( \overrightarrow{pq} = (1 - 2, 4 - 0, -3 - 2) = (-1, 4, -5) \)

Find the Parametric Equations

The parametric equations of a line through point P(2,0,2) with direction vector \( \overrightarrow{pq} = (-1, 4, -5) \) are given by \( x = x_1 + at \), \( y = y_1 + bt \) and \( z = z_1 + ct \) where \( (x_1, y_1, z_1) \) are the coordinates of the point, \( (a, b, c) \) are the components of the direction vector and \( t \) is the parameter. So the parametric equations of the line are \( x = 2 -t \), \( y = 4t \), \( z = 2 - 5t \)

Find the Symmetric Equations

The symmetric equations of the line can be found from the parametric equations by setting all equations equal to \( t \). This gives us \[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \] Substituting the known values, we get \( \frac{x - 2}{-1} = \frac{y}{4} = \frac{z - 2}{-5} \)

Key concepts

Direction Vector

A direction vector plays a pivotal role in defining the orientation of a line, especially in 3D space. When we have two distinct points, say point P and point Q, the direction vector, often denoted as \( \vec{pq} \), points from P to Q and can be easily obtained through vector subtraction.

In the context of our exercise, the direction vector is computed using the coordinates of points P (2,0,2) and Q (1,4,-3). By subtracting the coordinates of P from Q, \( \vec{pq} = Q - P = (1-2, 4-0, -3-2) \), we get the direction vector \( \vec{pq} = (-1, 4, -5) \). This vector not only provides a sense of direction but is also essential for writing the parametric and symmetric equations of the line.

Symmetric Equations

Symmetric equations provide a clever way to represent a line in 3D space without relying on a parameter. The essence of symmetric equations lies in their ability to equate the ratios of the differences between the coordinates to the components of the line's direction vector.

In the solved exercise, by eliminating the parameter \( t \) from the parametric equations and setting the ratios equal to each other, we derived the symmetric equations of the line as follows: \( \frac{x - 2}{-1} = \frac{y}{4} = \frac{z - 2}{-5} \). These symmetric equations are quite useful because they allow us to analyze the line's behavior and locate points on it in a directly proportional manner.

3D Coordinate Geometry

In the realm of 3D coordinate geometry, understanding spatial relationships between points, lines, and planes is essential. The coordinate system employs the \( x \) (horizontal), \( y \) (vertical), and \( z \) (depth) axes to define the position of points in space.

For our purposes, we described a line using the given points as anchors in this three-dimensional system. The spatial intuition provided by 3D geometry is critical when calculating quantities like the direction vector or when deriving parametric and symmetric equations.

Vector Subtraction

Vector subtraction is a fundamental operation used to find the direction vector in problems like these. It involves taking the corresponding components of one vector and subtracting them from another. If we consider vectors as arrows starting from the origin, subtracting one vector from another effectively places the tail of one arrow at the head of the other, pointing from the first point towards the second.

In our case, calculating \( \vec{pq} \) by subtracting point P from point Q, \( \vec{pq} = Q - P \) is a quintessential example of vector subtraction. This provides us with a new vector that encapsulates the direction and is necessary for further calculations in determining the line equation.