91Ó°ÊÓ

The parametric equation of a line in 3D space is a powerful way to express the infinite set of points that make up the line. Instead of describing a line using a standard algebraic equation, we use vectors.
This involves:

• A point on the line, known as the starting point or position vector, often represented as \( \mathbf{A} \) (here, \( (2,0,3) \)).
• A direction vector \( \mathbf{d} \) (here, \( (-3,0,-2) \)), which tells us the direction in which the line travels.
• A parameter, usually denoted \( t \), which scales the direction vector. It helps in finding all the points on the line as we change \( t \).

Hence, the parametric equation is written as \( \mathbf{r}(t) = \mathbf{A} + t\mathbf{d} \). By plugging in different values of \( t \), you generate various points \( \mathbf{r} \) along the line. This method is critical, especially in vector calculus, because it allows us to easily describe lines and curves in three-dimensional space.

The cross product is a vector operation useful in 3D vector calculus. Unlike the dot product, which results in a scalar, the cross product produces a vector perpendicular to the plane formed by the two original vectors.
To perform a cross product between two vectors \( \mathbf{a} \) and \( \mathbf{b} \), you use the determinant of a matrix composed of:

• The unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \),
• The components of \( \mathbf{a} \) in the second row,
• The components of \( \mathbf{b} \) in the third row.

The result is a new vector that is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \). This is particularly helpful in finding the shortest distance from a point to a line because the direction of the resulting vector directly correlates with the perpendicular to both input vectors.
For example, in this exercise, calculating the cross product of \( (\mathbf{P} - \mathbf{A}) \) and \( \mathbf{d} \) gives us a helpful perpendicular vector to measure the distance from a point to the line.

The distance formula in vector spaces is a versatile tool for measuring lengths between points and geometric entities like lines and planes. In 3D space, the formula is used to find the shortest distance of a point to a line.
The general formula given for the distance \( d \) from a point \( \mathbf{P} \) to a line defined by an equation \( \mathbf{r}(t) = \mathbf{A} + t\mathbf{d} \) is:
\[ d = \frac{\| (\mathbf{P} - \mathbf{A}) \times \mathbf{d} \|}{\| \mathbf{d} \|} \]
This formula covers essential steps:

• Finding the vector \( (\mathbf{P} - \mathbf{A}) \) from the point to any arbitrary point \( \mathbf{A} \) on the line.
• Calculating the cross product of this vector with the direction vector, yielding a vector orthogonal to both, pointing directly away from the line towards the point.
• Calculating the magnitude of this cross product, giving us the uphill climb from the line directly to the point.
• Finally, dividing by the magnitude of the direction vector to ensure we are measuring the true perpendicular distance.

This formula elegantly combines geometry and algebra, allowing us to measure real-world distances indicated as vectors in space.