Problem 20 - Thomas Calculus

Question

Express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k}.\) \(\overrightarrow{A B}\) if \(A\) is the point \((1,0,3)\) and \(B\) is the point \((-1,4,5)\)

Step-by-Step Solution

Answer

\( \overrightarrow{AB} = -2\mathbf{i} + 4\mathbf{j} + 2\mathbf{k} \)

Step 1: Identify the coordinates of the points

The coordinates of point \( A \) are given as \( (1, 0, 3) \) and those of point \( B \) are \( (-1, 4, 5) \). We will use these coordinates to calculate the vector \( \overrightarrow{AB} \).

Step 2: Subtract the coordinates of A from B

To find the vector \( \overrightarrow{AB} \), subtract the coordinates of point \( A \) from those of point \( B \). This is done component-wise: \( B_x - A_x \), \( B_y - A_y \), and \( B_z - A_z \).

Step 3: Calculate the components of the vector \( \overrightarrow{AB} \)

Calculate each component: - \( v_1 = B_x - A_x = -1 - 1 = -2 \) - \( v_2 = B_y - A_y = 4 - 0 = 4 \) - \( v_3 = B_z - A_z = 5 - 3 = 2 \)

Step 4: Express the vector in the desired form

Now that we have the components, express the vector \( \overrightarrow{AB} \) as:\[ \overrightarrow{AB} = -2\mathbf{i} + 4\mathbf{j} + 2\mathbf{k} \]

Key Concepts

Vector Subtraction

Vector subtraction is a fundamental operation in vector calculus. It allows us to find the difference between two vectors or effectively calculate the displacement direction from one point to another.
Imagine that vectors are arrows starting from an origin point. With vector subtraction, you find the new vector that connects these arrows' tips. In our exercise, we want to determine the vector from point \(A\) to point \(B\).
When subtracting vectors, you do it component-wise. Each axis (x, y, and z in 3D geometry) is considered separately. For the exercise, we computed component differences as follows:

• For the \(x\)-component: subtract the \(x\)-coordinate of \(A\) from the \(x\)-coordinate of \(B\).
• Repeat for the \(y\) and \(z\) components.

This step ensures that the resulting vector correctly represents the path from the origin point \((A)\) to the destination \((B)\). This technique is broadly applicable in physics, engineering, and computer graphics whenever you need to determine the relative position between points.

Coordinate Geometry

Coordinate geometry, sometimes called analytic geometry, allows us to represent geometric figures using algebraic equations and establishes a bridge between algebra and geometry. In this exercise, coordinate geometry helps us handle the three-dimensional space efficiently.
Every point in 3D space has three coordinates: \(x, y,\) and \(z\). These specify its location relative to the origin (0, 0, 0). By storing information this way, we can perform mathematical operations like addition, subtraction, and scalar multiplication with ease.
Here, points \(A\) and \(B\) were represented as \((1,0,3)\) and \((-1,4,5)\) respectively. These coordinates help us quickly compute the vector using subtraction.
This process illustrates how multi-dimensional data is manipulated, highlighting the powerful combination of geometry and algebra to solve complex problems like finding the shortest path or distance between points.

Vector Representation

Vector representation allows us to express spatial relationships in mathematical terms that are easy to manipulate. Vectors are often represented in physics and engineering using three components \(i, j, k\), which align with the three axes of 3D space.
Each component of a vector shows its magnitude along those axes. For example, in our exercise, the vector \(\overrightarrow{AB}\) translates to \(-2\mathbf{i} + 4\mathbf{j} + 2\mathbf{k}\).
This expression tells us:

• The vector moves "backwards" along the \(x\)-axis, noted by the negative \(-2\).
• "Upwards" by 4 units along the \(y\)-axis.
• "Forwards" by 2 units along the \(z\)-axis.

Such representation is not only crucial for visualizing vectors in space but it also simplifies complex calculations involving directions and magnitudes. This direct form of representation allows computations like dot products, cross products, and more in vector calculus.