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A vector equation is a succinct way to describe a line in three-dimensional space (R3). This type of equation combines information about a point on the line and its direction. To construct the vector equation of a line, you'll need two primary components:

• A point on the line, represented as a position vector, say \( \mathbf{A} = \begin{pmatrix} a_1 \ a_2 \ a_3 \end{pmatrix} \).
• A direction vector, which defines the line's direction, denoted by \( \mathbf{d} = \begin{pmatrix} d_1 \ d_2 \ d_3 \end{pmatrix} \).

With these, the vector equation is expressed as:\[\mathbf{r} = \mathbf{A} + t\mathbf{d} \quad (t \text{ is a scalar parameter})\]
This equation implies that any point \( \mathbf{r} \) on the line can be reached by starting at \( \mathbf{A} \) and moving in the direction of \( \mathbf{d} \) multiplied by a scalar \( t \). Varying \( t \) gives every point on the line, from negative to positive infinity.

The parametric form of a line in three-dimensional space offers a clear view of how each coordinate of the line depends on a single parameter. From the vector equation \( \mathbf{r} = \mathbf{A} + t\mathbf{d} \), where \( \mathbf{A} = \begin{pmatrix} a_1 \ a_2 \ a_3 \end{pmatrix} \) and \( \mathbf{d} = \begin{pmatrix} d_1 \ d_2 \ d_3 \end{pmatrix} \), the parametric form is derived by breaking down this vector equation into individual components:

• \( x = a_1 + td_1 \)
• \( y = a_2 + td_2 \)
• \( z = a_3 + td_3 \)

Each of these equations describes how the \( x \), \( y \), and \( z \) coordinates take values as the parameter \( t \) varies.
One of the main advantages of the parametric form is its ability to easily plug in different values for \( t \) to quickly find specific points along the line. This form of the equation highlights the linear relationship between the parameter \( t \) and the coordinates of the line.

Symmetric equations provide another way to represent lines in three-dimensional space, derived by eliminating the parameter \( t \) from the parametric equations. Given the parametric equations:

• \( x = a_1 + td_1 \)
• \( y = a_2 + td_2 \)
• \( z = a_3 + td_3 \)

The symmetric equations are obtained by solving for \( t \) in each parametric equation and equating them:
\[ \frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3} \]
These equations provide a direct method to express a line without explicitly using the parameter \( t \). Symmetric equations are particularly useful when you need to express the line's direction ratios directly, making it simple to verify if a point lies on the line or to compare directions of different lines quickly. They offer a holistic way to see all relationships between the coordinates of points on the line.