Problem 39 - Thomas Calculus

Question

If \(\overrightarrow{A B}=\mathbf{i}+4 \mathbf{j}-2 \mathbf{k}\) and \(B\) is the point \((5,1,3),\) find \(A.\)

Step-by-Step Solution

Answer

Point A is at (4, -3, 5).

Step 1: Understand the Position Vector Concept

To find a point given its position vector, we need to understand that the position vector \( \overrightarrow{AB} \) leads from point \( A \) to point \( B \). To backtrack to \( A \), we subtract the vector \( \overrightarrow{AB} \) from point \( B \).

Step 2: Write the Equation for Point A

Since \( \overrightarrow{AB} \) is given as \( \mathbf{i} + 4 \mathbf{j} - 2 \mathbf{k} \), and \( B \) is at \( (5,1,3) \), the coordinates of \( A \) are given by:\[ A = B - \overrightarrow{AB} \] Thus, we'll subtract the vector components from the coordinates of point \( B \).

Step 3: Perform the Subtraction

The subtraction is performed component-wise: \[ A_x = 5 - 1, \ A_y = 1 - 4, \ A_z = 3 - (-2) \] Which results in:\[ A_x = 4, \ A_y = -3, \ A_z = 5 \]

Step 4: Write the Coordinate of Point A

The coordinates of point \( A \) are thus:\[ A = (4, -3, 5) \].

Key Concepts

Position Vector

In the world of vector mathematics, a position vector is an essential concept. It is a vector that represents the position of a point in space relative to an origin. Imagine it as an arrow pointing from the origin to the point. In our example, the position vector \( \overrightarrow{AB} = \mathbf{i} + 4 \mathbf{j} - 2 \mathbf{k} \) anchors the concept of moving from point \( A \) to point \( B \).
To find point \( A \) when given \( B \) and the position vector \( \overrightarrow{AB} \), think of retracing the path backward. Subtract \( \overrightarrow{AB} \) from \( B \) to determine where point \( A \) lies.
Position vectors play a crucial role in coordinate geometry and physics, allowing us to describe positions and movements in a clear, mathematical manner.

• The position vector \( \overrightarrow{AB} \) means the direction and length from \( A \) to \( B \).
• To find the starting point (\( A \)), subtract the position vector from the endpoint (\( B \)).

Coordinate Geometry

Coordinate geometry, or analytic geometry, bridges the gap between algebra and geometry. It uses the Cartesian coordinate system to describe geometric figures algebraically. This system lets us pinpoint locations in space using coordinates (x, y, z).
In this exercise, we are given the coordinates of point \( B \) as \((5,1,3)\). The task is to find point \( A \), using the concept of a position vector. The vector \( \overrightarrow{AB} \) shows how point \( B \) is related to point \( A \), through coordinate geometry.
We use coordinate geometry to solve for point \( A \) by applying the rule of vector subtraction. Ultimately, it ties together the location of points with vector directions using the familiar plane of coordinates.

• Coordinates tell the specific location of a point in space.
• The process involves finding the unknown coordinates by using vector relationships.

Vector Subtraction

Vector subtraction is a fundamental operation in vector mathematics. It is as straightforward as subtracting vectors component by component. When we deal with vectors such as \( \overrightarrow{AB} \), subtracting allows us to transition between points effectively.

In our task, to find the coordinates of point \( A \), we use the vector subtraction process. We have point \( B \) at \((5,1,3)\) and \( \overrightarrow{AB} \) as \( \mathbf{i} + 4 \mathbf{j} - 2 \mathbf{k} \). Thus, to perform subtraction, we follow these steps:

• Subtract the \( x \) component: \( A_x = 5 - 1 \).
• Subtract the \( y \) component: \( A_y = 1 - 4 \).
• Subtract the \( z \) component: \( A_z = 3 - (-2) \).

This results in the coordinates of \( A \) as \((4, -3, 5)\). Each component was calculated individually, making vector subtraction a systematic and easy-to-follow procedure.
By understanding and applying vector subtraction, you can solve similar problems involving the movement or transformation of points in space.