Problem 2 - Advanced Engineering Mathematics

Question

In Problems, find a vector equation for the line through the given points. $$ (0,4,5),(-2,6,3) $$

Step-by-Step Solution

Answer

The vector equation is \( \mathbf{r}(t) = (0,4,5) + t(-2,2,-2) \).

Step 1: Identify Given Points

We are given two points, \((0, 4, 5)\) and \((-2, 6, 3)\). A vector equation for a line can be formed using these points.

Step 2: Determine the Direction Vector

The direction vector \( \mathbf{d} \) is found by subtracting the coordinates of the first point from the second point. Compute \((-2, 6, 3) - (0, 4, 5)\) to get \( \mathbf{d} = (-2, 2, -2) \).

Step 3: Construct the Vector Equation of the Line

Use the vector equation formula \( \mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{d} \), where \( \mathbf{r}_0 \) is the position vector of a point on the line (choose \( (0,4,5) \)) and \( \mathbf{d} = (-2,2,-2) \). Thus, the vector equation is \( \mathbf{r}(t) = (0, 4, 5) + t(-2, 2, -2) \).

Step 4: Express the Equation in Component Form

Substitute the expressions into the equation: \( \mathbf{r}(t) = (0 - 2t, 4 + 2t, 5 - 2t) \). This presents the line in component form describing every point on the line as a function of parameter \( t \).

Key Concepts

Direction Vector

When discussing a line in vector form, one of the key components you need to comprehend is the **direction vector**. This vector is essentially like an arrow that shows how the line extends into space. It plays a crucial role in determining the orientation or "direction" of your line.

To find the direction vector, use vector subtraction. Suppose you are given two points in space, say Point A \(x_1, y_1, z_1\) and Point B \(x_2, y_2, z_2\). The direction vector \(\mathbf{d}\) is simply \(\mathbf{d} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)\).

• It helps locate every other point on the line by traveling further in this direction.
• In linear algebra, the direction vector is vital because it helps define lines and their equations in vector form.

Position Vector

Another major player in the vector equation of a line is the **position vector**. This is essentially a vector that points to a specific location in space, starting from the origin (0, 0, 0).

In a vector equation \(\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{d}\), the position vector \(\mathbf{r}_0\) is the initial point of the line.

• Think of the position vector as the place where you start drawing your line.
• It's crucial for setting up the vector equation of a line because it acts as the anchor point.

For instance, if you're working with the point (0, 4, 5), then \(\mathbf{r}_0 = (0, 4, 5)\) as a position vector points directly to this place in space, functioning as a sort of reference for the entire line.

Vector Subtraction

The process of finding the difference between two vectors is called **vector subtraction**. This operation is pivotal not just in finding direction vectors, but also in understanding how distances and displacements work in vector spaces.

To perform vector subtraction, you simply subtract each corresponding component of the vectors. This means for vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), the difference is \(\mathbf{a} - \mathbf{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3)\).

• Vector subtraction allows you to find the displacement between two points.
• It serves as the foundational operation to calculate direction vectors.

Understanding vector subtraction is crucial in linear algebra and physics, as it enables you to measure how one vector changes relative to another.

Linear Algebra

**Linear Algebra** is the branch of mathematics dealing with vectors and functions used to manipulate these vectors. It provides the framework for many techniques in mathematics, including the study and formulation of vector equations of lines.

In the context of a vector equation for a line, linear algebra involves principles like vector addition, scalar multiplication, and understanding spaces and transformations.

• It helps describe and interpret spaces where vector lines exist, offering the tools to solve real-world problems.
• Linear algebra models problems through systems of vectors, making complex relationships easier to visualize and solve.

Overall, linear algebra supports the vector equation of a line by providing the mathematical grammar needed to express vectors and transformations in different spaces.