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: Understand the Problem

We need to find the coordinates of point B given vector \( \overrightarrow{A B} \) and point A. The vector gives us the direction and distance from point A to point B.

Step 2: Set Up the Equation for Vector Addition

The position vector of point B can be found by adding vector \( \overrightarrow{A B} \) to point A's coordinates. If point A is represented as \( A(x_1, y_1, z_1) \) and vector \( \overrightarrow{AB} \) has components \( \langle a, b, c \rangle \), then the coordinates of point B \( B(x_2, y_2, z_2) \) are given by: \( x_2 = x_1 + a \), \( y_2 = y_1 + b \), \( z_2 = z_1 + c \).

Step 3: Substitute Given Values

Point A is given as \((-2, -3, 6)\) and vector \( \overrightarrow{A B} \) is \(-7 \mathbf{i} + 3 \mathbf{j} + 8 \mathbf{k} \). Substitute these into the formulas: \( x_2 = -2 - 7 \), \( y_2 = -3 + 3 \), \( z_2 = 6 + 8 \).

Step 4: Calculate the Coordinates of B

Perform the calculations to find the coordinates of point B:\( x_2 = -2 - 7 = -9 \),\( y_2 = -3 + 3 = 0 \),\( z_2 = 6 + 8 = 14 \).

Step 5: Formulate the Final Answer

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

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of geometry where we use a coordinate system to define and represent points, lines, and shapes. In three dimensions, each point is represented by a set of numerical coordinates: \((x, y, z)\). These coordinates can be thought of as stretches or travels from the origin along the x, y, and z axes, respectively.

In this exercise, point A has coordinates \((-2, -3, 6)\). When given a vector like \(\overrightarrow{AB} = -7 \, \mathbf{i} + 3 \, \mathbf{j} + 8 \, \mathbf{k}\), it describes a direction and distance from point A to point B in the coordinate system.

• The vector's \(\mathbf{i}\), \((\mathbf{j})\), and \((\mathbf{k})\) components lie along the x, y, and z directions.
• The goal is to move from point A to point B using this displacement vector.

Understanding coordinate geometry helps us visualize complex spatial problems more clearly and solve them systematically.

Vector Components

Vectors are fundamental in physics and mathematics as they offer a mathematical representation of physical quantities with both magnitude and direction. For a vector, the components are crucial as they individually represent the influence or magnitude of the vector in each of the coordinate directions.

A vector such as \(\overrightarrow{AB} = -7 \, \mathbf{i} + 3 \, \mathbf{j} + 8 \, \mathbf{k}\) shows its components in the x, y, and z directions. We can interpret this as:

• \

3D Vectors

3D vectors extend the concept of 2D vectors into three dimensions. These vectors have three components, corresponding to the x, y, and z axes in space. They are essential for describing spatial movements and forces, such as the one shown in this problem.

• A 3D vector like \(\overrightarrow{AB} = -7 \, \mathbf{i} + 3 \, \mathbf{j} + 8 \, \mathbf{k}\) gives a full description of movement in three directions simultaneously.
• This allows us to calculate new positions in 3D coordinates effectively, such as moving from point A's position by this given vector to deduce point B.

By using these components, we can ascertain any displacement, directionality, or full spacial picture in a 3D context. Understanding 3D vectors is critical for any problem involving movement or location changes in space.