Problem 1 - Advanced Engineering Mathematics

Question

In Problems, find a vector equation for the line through the given points. $$ (1,2,1),(3,5,-2) $$

Step-by-Step Solution

Answer

The vector equation is \( \mathbf{r} = (1, 2, 1) + t(2, 3, -3) \).

Step 1: Identify Points

The problem provides two points through which the line passes: \( P_1 = (1, 2, 1) \) and \( P_2 = (3, 5, -2) \).

Step 2: Find the Direction Vector

Calculate the direction vector \( \mathbf{d} \) by subtracting the coordinates of \( P_1 \) from \( P_2 \): \( \mathbf{d} = (3-1, 5-2, -2-1) = (2, 3, -3) \).

Step 3: Create the Vector Equation

The vector equation of a line can be written as \( \mathbf{r} = \mathbf{r_0} + t\mathbf{d} \), where \( \mathbf{r_0} \) is a position vector to a point on the line (we can use \( P_1 = (1, 2, 1) \)) and \( \mathbf{d} \) is the direction vector. Thus, the vector equation becomes \( \mathbf{r} = (1, 2, 1) + t(2, 3, -3) \).

Key Concepts

Direction Vector

When dealing with lines in three-dimensional space, the direction vector is a central concept. It essentially tells us the direction in which the line extends. To find this vector, we use two points on the line. Consider the points given in the exercise: \( P_1 = (1, 2, 1) \) and \( P_2 = (3, 5, -2) \). The direction vector \( \mathbf{d} \) is obtained by subtracting the coordinates of \( P_1 \) from \( P_2 \):

• This gives us \( \mathbf{d} = (3-1, 5-2, -2-1) = (2, 3, -3) \).

This vector \( \mathbf{d} = (2, 3, -3) \) effectively captures the line's direction, providing us with a sense of both the angle and orientation in three-dimensional space.
The beauty of the direction vector is its simplicity. Regardless of where you start on the line, multiplying this vector by any scalar \( t \) will point you somewhere else along the line. The line literally stretches in the direction dictated by \( \mathbf{d} \). It helps create the vector equation of the line in three-dimensional geometry.

Position Vector

The position vector is our anchor point in describing a line in space. It represents a specific point on the line, usually one of the provided points. For making things simple and straight, we often choose the first point given in an exercise.
In the case of the given exercise, we have: \( P_1 = (1, 2, 1) \). The resulting position vector, \( \mathbf{r_0} = (1, 2, 1) \), gives us a starting point or a fixed location on the line.

• It ensures that our line equation is grounded on an actual point in space rather than floating unpredictably.
• Crucially, the position vector defines a direction from the origin \((0, 0, 0)\) to this point \((1, 2, 1)\).

Together with a direction vector, the position vector turns abstract algebra into physical intuition. We plug this known point into the vector equation of the line, ensuring that as we navigate the line by varying \( t \), we always have a "home base" to tether our line visually and mathematically.

Three-Dimensional Coordinate Geometry

Three-dimensional coordinate geometry allows us to explore spaces beyond the flat confines of two-dimensional planes. When plotting lines in this space, we describe them using vector equations. This is a standard approach, as vectors easily translate movements and directions across three axes.
In three-dimensional geometry, each point is specified using three coordinates \((x, y, z)\). For example, the points \((1, 2, 1)\) and \((3, 5, -2)\) from the exercise represent specific locations in space.

• The vector equation \( \mathbf{r} = \mathbf{r_0} + t\mathbf{d} \) conveniently represents a line.
• Here, \( \mathbf{r_0} \) is our position vector, and \( \mathbf{d} \) is our direction vector.
• By using a parameter \( t \), the line can extend infinitely in both directions.

This modular form equations mean a line can stretch continuously across space. We can adjust \( t \) to see where on the line we end up. Three-dimensional coordinate geometry provides a comprehensive framework to analyze complex spatial relationships, which is pivotal for fields like physics, engineering, and computer graphics.