Problem 40 - Thomas Calculus

Question

If \(\overrightarrow{A B}=-7 \mathbf{i}+3 \mathbf{j}+8 \mathbf{k}\) and \(A\) is the point \((-2,-3,6),\) find \(B\)

Step-by-Step Solution

Answer

Point B is at (-9, 0, 14).

Step 1: Understanding Vectors

The vector \(\overrightarrow{AB}\) describes the direction and distance from point \(A\) to point \(B\). Given vector \(\overrightarrow{AB}=-7 \mathbf{i}+3 \mathbf{j}+8 \mathbf{k}\) means we need to move from point \(A\) by \(-7\) units in the direction of the x-axis, \(+3\) units in the direction of the y-axis, and \(+8\) units in the direction of the z-axis.

Step 2: Writing the Position of Point B

If \(A\) is the point \((-2,-3,6)\), then \(B\) can be found by applying the components of the vector \(\overrightarrow{AB}\) to the coordinates of point \(A\).

Step 3: Calculating Coordinate for x-axis

The x-coordinate of \(B\) is calculated by adding the x-component of \(\overrightarrow{AB}\) to the x-coordinate of \(A\): \(-2 - 7 = -9\).

Step 4: Calculating Coordinate for y-axis

The y-coordinate of \(B\) is found by adding the y-component of \(\overrightarrow{AB}\) to the y-coordinate of \(A\): \(-3 + 3 = 0\).

Step 5: Calculating Coordinate for z-axis

The z-coordinate of \(B\) is determined by adding the z-component of \(\overrightarrow{AB}\) to the z-coordinate of \(A\): \(6 + 8 = 14\).

Step 6: Solution

Thus, the coordinates of point \(B\) are \((-9, 0, 14)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebra to study geometric problems. It involves the use of coordinates to represent points, lines, and figures in a given plane or space. In simple terms, it allows us to describe geometric shapes with algebraic equations.

In 3D space, any point is represented by three coordinates:

• The x-coordinate, which indicates the position along the horizontal axis (left to right).
• The y-coordinate, which represents the position along the vertical axis (up and down).
• The z-coordinate, which specifies the position in depth (front to back).

These coordinates allow us to pinpoint any location within 3D space. By applying coordinate geometry, we can solve various spatial problems, such as finding the distance between two points, determining the midpoint of a segment, or, as in our exercise, finding a new point when given a direction and distance to move from another point.

3D Vectors

3D vectors are mathematical entities used to represent quantities that have both magnitude and direction in three-dimensional space. A vector can be thought of as an arrow that points from one location to another, having a certain length (magnitude) and pointing in a specific way (direction).

Vectors are often represented using components along the x, y, and z axes. For instance, \[ \overrightarrow{AB} = -7 \mathbf{i} + 3 \mathbf{j} + 8 \mathbf{k} \]This describes a vector that moves:

• -7 units in the x-direction
• +3 units in the y-direction
• +8 units in the z-direction

Understanding vectors is crucial in 3D geometry because they provide a simple way to describe translations, rotations, and various physical phenomena such as force and velocity. They help us solve problems involving displacement like in the provided exercise, where the vector describes how to get from point A to point B.

Displacement Vector

A displacement vector is a type of vector that represents a change in position in space. It tells us how to get from one point to another. In the context of coordinate geometry and 3D spaces, displacement vectors are especially useful to find new positions based on a given starting point and direction.

For example, the vector \[ \overrightarrow{AB} = -7 \mathbf{i} + 3 \mathbf{j} + 8 \mathbf{k} \]acts as a set of instructions to move from point A, given as \((-2, -3, 6)\), by -7 units along the x-axis, +3 units along the y-axis, and +8 units along the z-axis.

To find point B, apply these instructions to the coordinates of point A. Calculation steps are:

• Add the x-component of the vector to the x-coordinate of point A: \(-2 - 7 = -9\text{ for the x-coordinate}\).
• Add the y-component of the vector to the y-coordinate of point A: \(-3 + 3 = 0\text{ for the y-coordinate}\).
• Add the z-component of the vector to the z-coordinate of point A: \(6 + 8 = 14\text{ for the z-coordinate}\).

As a result, the displacement gives us the final position of point B: \((-9, 0, 14)\). Displacement vectors simplify finding new points by effectively using vectors to manage direction and movement in 3D coordinate systems.