Problem 18 - University Calculus: Early Transcendentals

Question

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(\overrightarrow{P_{1} P_{2}}\) if \(P_{1}\) is the point (1,2,0) and \(P_{2}\) is the point (-3,0,5)

Step-by-Step Solution

Answer

The vector is \(-4 \mathbf{i} - 2 \mathbf{j} + 5 \mathbf{k}\).

Step 1: Identify the coordinates

To express the vector \( \overrightarrow{P_{1} P_{2}} \) in the form \( \mathbf{w}=w_{1} \mathbf{i}+w_{2} \mathbf{j}+w_{3} \mathbf{k} \), we first identify the coordinates of points \( P_1 \) and \( P_2 \).\( P_1 = (1, 2, 0) \) and \( P_2 = (-3, 0, 5) \).

Step 2: Calculate the vector components

The components of the vector \( \overrightarrow{P_1 P_2} \) can be found by subtracting the coordinates of \( P_1 \) from \( P_2 \).\( w_1 = -3 - 1 \), \( w_2 = 0 - 2 \), \( w_3 = 5 - 0 \).This results in components: \( w_1 = -4 \), \( w_2 = -2 \), \( w_3 = 5 \).

Step 3: Express the vector in terms of unit vectors

Now, we express \( \overrightarrow{P_1 P_2} \) using the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).Given the components, the vector \( \overrightarrow{P_1 P_2} = -4 \mathbf{i} - 2 \mathbf{j} + 5 \mathbf{k} \).

Key Concepts

Vector Components

Vectors are mathematical objects that have both a magnitude and a direction. Each vector in three-dimensional space can be expressed through its components along the x, y, and z axes. These are referred to as the vector's components. To find the components of a vector from one point to another, such as \( \overrightarrow{P_1 P_2} \), you subtract the coordinates of the initial point \( P_1 \) from those of the terminal point \( P_2 \). This gives the change in each direction.

• The x-component is found by subtracting the x-coordinate of \( P_1 \) from the x-coordinate of \( P_2 \).
• The y-component is found similarly, using the y-coordinates.
• The z-component is found by subtracting z-coordinates.

For example, if \( P_1 = (1, 2, 0) \) and \( P_2 = (-3, 0, 5) \), the vector components are:\[ w_1 = -3 - 1 = -4 \]\[ w_2 = 0 - 2 = -2 \]\[ w_3 = 5 - 0 = 5 \]These represent how far and in what direction the vector stretches in each of the three dimensions.

Unit Vectors

After finding the components of a vector, it is often useful to express the vector using unit vectors. Unit vectors are vectors of length one that point in the direction of the coordinate axes. These vectors are represented as \( \mathbf{i}, \mathbf{j}, \text{and } \mathbf{k} \).

• \( \mathbf{i} \) is the unit vector in the direction of the x-axis.
• \( \mathbf{j} \) is the unit vector in the direction of the y-axis.
• \( \mathbf{k} \) is the unit vector in the direction of the z-axis.

To express a vector using unit vectors, you multiply each component of the vector by the corresponding unit vector. This method makes it simple to visualize the vector direction and calculate it mathematically.For the vector \( \overrightarrow{P_1 P_2} = -4 \mathbf{i} - 2 \mathbf{j} + 5 \mathbf{k} \), each term shows the magnitude of the vector in the direction of \( \mathbf{i}, \mathbf{j}, \text{and } \mathbf{k} \) respectively. This form is incredibly useful in vector calculus for notation and calculation.

Point Coordinates

Understanding point coordinates is essential when working with vectors in vector calculus. Coordinates are numerical representations of any location in a space. In three-dimensional space, coordinates are expressed as \( (x, y, z) \), indicating a point's position.When you have two points, say \( P_1 = (1, 2, 0) \) and \( P_2 = (-3, 0, 5) \), you understand exactly where each point is in relation to the origin. The origin \( (0, 0, 0) \) is the point where all three axes meet, often considered the starting point for any measurement in 3D space.Knowing the coordinates of two points allows you to calculate vectors that describe direction and distance between these points. This calculative process involves subtracting the coordinates of the first point from the second, to find how much you need to "move" in each direction to get from the first point to the second. Coordinates thus provide not only location but also a basis for creating vectors.