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A vector equation is a powerful way to represent a line in three-dimensional space. It combines both direction and a specific point the line passes through into a unified expression. In essence, the vector equation of a line expresses every point on the line in terms of vectors. The formula for a line's vector equation is given by:\[ \mathbf{r}(t) = \mathbf{r_0} + t\mathbf{v} \]where:

• \(\mathbf{r}(t)\) is the position vector of any point on the line as a function of parameter \(t\).
• \(\mathbf{r_0}\) is a fixed point on the line.
• \(\mathbf{v}\) is the direction vector, showing how the line extends from the point \(\mathbf{r_0}\).
• \(t\) is a scalar parameter that can take any real number value.

Using the vector equation helps visualize the line's extent in any direction, as the parameter \(t\) scales the direction vector, allowing the position vector \(\mathbf{r}(t)\) to trace every point on the line.
In this exercise, the vector equation becomes:\[ \mathbf{r}(t) = \langle 6, -5, 2 \rangle + t \langle 1, 3, -\frac{2}{3} \rangle \].

Parametric equations express the coordinates of the points on the line as separate equations, each dependent on a parameter, often denoted as \(t\). From the vector equation, you can derive the parametric form by identifying each coordinate separately.To convert the vector equation given by \(\mathbf{r}(t) = \langle 6, -5, 2 \rangle + t \langle 1, 3, -\frac{2}{3} \rangle\), we break it down into its component equations:

• For the x-coordinate: \(x(t) = 6 + t\)
• For the y-coordinate: \(y(t) = -5 + 3t\)
• For the z-coordinate: \(z(t) = 2 - \frac{2}{3}t\)

Each equation provides how each coordinate changes as \(t\) changes. This representation is handy in applications like computer graphics and modeling motion, where the variations of coordinates over time or another variable are important to analyze.
Parametric equations keep the line's nature flexible, allowing various values of \(t\) to map out every point on the line.

The direction vector defines the course that a line takes through space. It is essential because it dictates how the line extends in 3D space from a fixed starting point. In the vector equation, the direction vector is noted as \(\mathbf{v}\).The direction vector is non-zero and usually derived from known points or directly given in problems involving lines. In our example, \(\mathbf{v} = \langle 1, 3, -\frac{2}{3} \rangle\), indicating the line moves 1 unit in the x-direction, 3 units in the y-direction, and \(-\frac{2}{3}\) units in the z-direction.

• By scaling the direction vector with different values of \(t\), each point on the line can be reached.
• The direction vector does not change its initial orientation with scaling but affects how far along the line you travel.

Understanding the direction vector is critical in geometry, physics, and vector calculus, where understanding the direction of lines relative to planes or other vectors can offer insights into spatial relationships.

To fully define a line in space, a specific point on the line is needed. This point acts as an anchor in the vector equation. It’s usually one of the given elements when forming the equation of a line.In the given exercise, the point \((6, -5, 2)\) is used in the vector equation:\[ \mathbf{r_0} = \langle 6, -5, 2 \rangle \]

• This point is essential as it provides a fixed position from which the line is extended in the direction of the direction vector.
• It also ensures that any line equation is unique to that specific line through that point.

The point in 3D space helps ground our understanding of geometric lines. Applying a point on the line with a direction offers a complete description of the line’s position and path through the vector and parametric equations. This concept is crucial when determining equations of lines and visualizing them spatially.