Problem 20 - University Calculus: Early Transcendentals

Question

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(\overrightarrow{A B}\) 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 Coordinates

The given points are \( A(1,0,3) \) and \( B(-1,4,5) \). We will use these to determine the vector \( \overrightarrow{AB} \).

Step 2: Calculate Vector Components

To find the vector \( \overrightarrow{AB} \), subtract the coordinates of point \( A \) from the coordinates of point \( B \). The calculation will be: \[\overrightarrow{AB} = (x_B - x_A, y_B - y_A, z_B - z_A)\]\[\overrightarrow{AB} = (-1 - 1, 4 - 0, 5 - 3)\]This results in \( \overrightarrow{AB} = (-2, 4, 2) \).

Step 3: Write Vector in Component Form

Express the vector \( \overrightarrow{AB} \) in the form of \( \mathbf{w} = w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k} \), using the components found in the previous step. So, \( \overrightarrow{AB} = -2 \mathbf{i} + 4 \mathbf{j} + 2 \mathbf{k} \).

Key Concepts

Vector Components

Vectors are directional quantities often represented in terms of their components along specific axes. Think of components as pieces that make up the vector. Each component shows how far the vector extends in the direction of each axis.
For three-dimensional vectors like \(\overrightarrow{AB} = -2 \mathbf{i} + 4 \mathbf{j} + 2 \mathbf{k}\), each component corresponds to a fundamental unit vector (\(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\)) along the x, y, and z axes respectively.

• \(w_1 = -2\) represents the displacement along the x-axis.
• \(w_2 = 4\) represents the displacement along the y-axis.
• \(w_3 = 2\) represents the displacement along the z-axis.

By combining these vector components, we gain a full picture of the vector's direction and magnitude.

Coordinate Points

Coordinate points are used to precisely locate a position in space. Each point consists of values that designate its location along the x, y, and z axes. For instance, point \(A(1,0,3)\) indicates that the object is positioned at \(1\) unit along the x-axis, \(0\) unit along the y-axis, and \(3\) units along the z-axis.
Coordinate points are fundamental because they provide a clear reference for determining the position and displacement of vectors. By marking the start \(A(1,0,3)\) and end \(B(-1,4,5)\) points of a vector, we can analyze the initial and final positions. This allows us to calculate the vector displacement or directional change by subtracting the coordinates of these points. Remember to write each coordinate clearly; they serve as the foundation for further vector operations.

Vector Subtraction

Vector subtraction is an essential operation used to find the difference between two vectors or points. The result showcases the change or displacement from the initial to the final point.
To perform vector subtraction, follow these steps:

• Identify the coordinates of both points involved. For example, with points \(A(1,0,3)\) and \(B(-1,4,5)\).
• Subtract corresponding coordinates:
  • \(x_B - x_A = -1 - 1 = -2\)
  • \(y_B - y_A = 4 - 0 = 4\)
  • \(z_B - z_A = 5 - 3 = 2\)
• Combine these results to form the vector \(\overrightarrow{AB} = (-2, 4, 2)\).

This process translates into the vector's representation expressed using unit vectors, ultimately giving more practical insight into spatial changes.