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To define a line in three-dimensional space, we start by determining its direction vector. The direction vector indicates the line's orientation and provides a unique identifier for its direction. To find this vector, we take two points known to lie on the line. For this specific example, the points are given as \(A(0,0,0)\) and \(B(1,2,3)\).

To calculate the direction vector, subtract the coordinates of point \(A\) from point \(B\):

• For the x-component: \((1 - 0) = 1\)
• For the y-component: \((2 - 0) = 2\)
• For the z-component: \((3 - 0) = 3\)

Thus, the direction vector \(\mathbf{d}\) is \((1, 2, 3)\).

This step is crucial because it not only shows the line's direction but helps in forming the vector and parametric equations of the line.

Once the direction vector is determined, the next step is to express the line using parametric equations. These equations describe each coordinate (x, y, z) as a function of a parameter \(t\), allowing for easy computation of any point along the line.

The parametric equations are derived from the vector equation of the line. Given the vector equation:\[ \mathbf{r}(t) = \mathbf{r}_0 + t \cdot \mathbf{d} \]where \(\mathbf{r}_0 = (0,0,0)\) (point \(A\)) and \(\mathbf{d} = (1,2,3)\) (the direction vector), we establish:

• For the x-coordinate: \(x(t) = 0 + 1 \cdot t = t\)
• For the y-coordinate: \(y(t) = 0 + 2 \cdot t = 2t\)
• For the z-coordinate: \(z(t) = 0 + 3 \cdot t = 3t\)

The resulting parametric equations make it simple to find specific points on the line just by plugging in different values of \(t\). This form is particularly useful for rendering the trajectory of the line in three-dimensional modeling.

In three-dimensional geometry, representing a line requires understanding its full equation, which combines direction and points. The 3D line equation uses vector and parametric representations to precisely convey spatial position.

Consider the vector equation obtained:\[ \mathbf{r}(t) = (0,0,0) + t \cdot (1,2,3)\]From this, we derive the parametric equations, succinctly giving us a line that can be mathematically navigated in 3D space:

• \(x = t\)
• \(y = 2t\)
• \(z = 3t\)

Each representation of the line provides a different level of interaction, whether plugging into computations or directly observing in an environment. Understanding these line equations encourages modeling accuracy and enhances spatial reasoning in 3D space.