In the following exercises, points P and Q are given. Let L...

In the following exercises, points $P$ and $Q$ are given. Let $L$ be the line passing through points $P$ and $Q .$ a. Find the vector equation of line $L$ b. Find parametric equations of line $L$ c. Find symmetric equations of line $L$ d. Find parametric equations of the line segment determined by $P$ and $Q$ . $P(4,0,5), Q(2,3,1)$

In the following exercises, points $P$ and $Q$ are given. Let $L$ be the line passing through points $P$ and $Q .$
a. Find the vector equation of line $L$
b. Find parametric equations of line $L$
c. Find symmetric equations of line $L$
d. Find parametric equations of the line segment determined by $P$ and $Q$ . $P(4,0,5), Q(2,3,1)$

Key Concepts

Vector Equation of a Line

This concept involves expressing a line in space using a fixed point on the line and a direction vector that indicates the line's direction. The vector equation is typically written as r = r0 + tv, where r0 is the position vector of a known point on the line, v is the direction vector obtained by subtracting the coordinates of two distinct points on the line, and t is a scalar parameter. This form is fundamental in analytic geometry and vector calculus for representing lines in any dimension.

Parametric Equations

Parametric equations express each coordinate of points on the line as functions of a single parameter. Derived from the vector equation, they represent the line by breaking the position vector into its components. Each coordinate (x, y, z) is written in the form x = x0 + tv_x, y = y0 + tv_y, and z = z0 + tv_z. This form is particularly useful for understanding and plotting the trajectory of points along the line.

Symmetric Equations

Symmetric equations are obtained by eliminating the parameter from the set of parametric equations of a line, provided that none of the direction vector components are zero. The result is an equation that relates the coordinates directly, usually presented as (x - x0)/v_x = (y - y0)/v_y = (z - z0)/v_z. This provides a concise and parameter-free representation of the line, emphasizing the inherent proportional relationships between the coordinates along the line.

Line Segment Representation

This concept involves constraining the parameter in the vector or parametric equations to a closed interval in order to represent only the portion of the line between two specific points. By choosing a parameter range, typically from 0 to 1, the resulting equations yield the line segment directly connecting the endpoints, reflecting the idea of interpolation between the points and ensuring that only the points on the segment are included.

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