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In vector calculus, the vector equation of a line is a powerful way to describe a line in terms of vectors. It expresses the position of any point on the line using a mathematical formula. The basic form of a vector equation of a line passing through a point \( P \) and parallel to a direction vector \( \vec{d} \) is given by:

• \( \vec{r}(t) = \vec{r}_0 + t\vec{d} \)

where:

• \( \vec{r}(t) \) is the vector representing a point on the line.
• \( \vec{r}_0 \) is the position vector of the point \( P \).
• \( t \) is a scalar parameter, representing how far to move along the line.
• \( \vec{d} \) is the direction vector indicating the line’s direction.

For this exercise, with \( P = (1,1) \) and \( \vec{d} = \langle 2,3 \rangle \), we substitute into the vector equation to get:

• \( \vec{r}(t) = \langle 1,1 \rangle + t\langle 2,3 \rangle \)
• Which simplifies to \( \vec{r}(t) = \langle 1+2t, 1+3t \rangle \)

This formula gives us a clear and concise way to see how the line is formed by moving along the direction vector from the initial point.

Parametric equations are derived from the vector equation and represent a line using variables linked to a parameter, typically \( t \). These equations express each coordinate of a point on the line as a separate function of the parameter.From the vector equation \( \vec{r}(t) = \langle 1+2t, 1+3t \rangle \), we can see how the line's coordinates develop independently:

• The parametric equation for \( x \) is: \( x = 1 + 2t \)
• Similarly, for \( y \): \( y = 1 + 3t \)

In these equations:

• \( t \) controls the position along the line.
• As \( t \) changes, \( (x, y) \) traces all points on the line.

These parametric forms are particularly useful in computational applications, such as computer graphics and physics simulations, because they provide a convenient means to iterate over points on a line.

Symmetric equations eliminate the parameter entirely, making them particularly useful for various mathematical analysis scenarios. They express the relationship between \( x \), \( y \) (and \( z \), if in 3D) directly, without a parameter, and are often used to determine if a point lies on a given line.We derive the symmetric equations from the parametric versions \( x = 1 + 2t \) and \( y = 1 + 3t \). By solving for \( t \) in each:

• From \( x = 1 + 2t \), solve for \( t \): \( t = \frac{x-1}{2} \)
• From \( y = 1 + 3t \), solve for \( t \): \( t = \frac{y-1}{3} \)

Setting these equal for consistency yields the symmetric equation:

• \( \frac{x-1}{2} = \frac{y-1}{3} \)

This form is valuable in quickly determining the specific characteristics of the line's alignment. It provides a straightforward check if a certain point \((x, y)\) is on the line, by verifying this equation holds true.