Problem 12 - Essential Calculus Early Transcendentals

Question

Find the sum of the given vectors and illustrate geometrically. $$\langle 1,3,-2\rangle, \quad\langle 0,0,6\rangle$$

Step-by-Step Solution

Answer

The sum of the vectors is \(\langle 1, 3, 4 \rangle\).

Step 1: Understand the Vectors

We have two vectors given: \( \langle 1, 3, -2 \rangle \) and \( \langle 0, 0, 6 \rangle \). Our task is to find the sum of these vectors both algebraically and geometrically.

Step 2: Adding the Vectors Algebraically

To add vectors, we add their corresponding components. So, we add the x-components, y-components, and z-components separately:\[\langle 1, 3, -2 \rangle + \langle 0, 0, 6 \rangle = \langle 1 + 0, 3 + 0, -2 + 6 \rangle\]This simplifies to:\[\langle 1, 3, 4 \rangle\]

Step 3: Illustrate Geometrically

To illustrate this geometrically, think of vector addition as placing the tail of the second vector at the tip of the first vector. The vector sum \( \langle 1, 3, 4 \rangle \) can be visualized as the diagonal of the parallelogram (or triangle in three-space) formed with the given vectors. This vector can be drawn in a 3D coordinate system starting from the origin traveling to the point \( (1, 3, 4) \).

Step 4: Verify and Interpret the Result

Re-check the addition of components to ensure accuracy: The x-component remains 1, y-component is 3, and the z-component is calculated as \(-2 + 6 = 4\). Hence, the sum vector \( \langle 1, 3, 4 \rangle \) is correct and represents a new position or direction that effectively combines the influence of both vectors in three-dimensional space.

Key Concepts

3D Coordinate System

In the 3D coordinate system, we deal with three axes: the x-axis, the y-axis, and the z-axis. Each of these axes represents a different dimension of space. By combining these three dimensions, we can accurately describe the position of any point in three-dimensional space.

A vector in this system can be described using a set of three numbers, correctly known as its components. These components are written in the form of a triplet, such as \( \langle x, y, z \rangle \). Here, the values \( x \), \( y \), and \( z \) correspond to the vector's position along the x, y, and z axes, respectively.

• The x-component determines the distance along the horizontal plane.
• The y-component determines the distance along the vertical plane.
• The z-component determines the distance vertically in terms of depth (into and out of the page or screen).

Understanding these components is crucial when adding vectors or interpreting their direction in space.

Geometric Interpretation

Visualizing vector addition can help clarify how vectors combine in three-dimensional space. When you add two vectors, such as \( \langle 1, 3, -2 \rangle \) and \( \langle 0, 0, 6 \rangle \), you are essentially performing a geometric operation.

Imagine placing the tail of the second vector at the head of the first vector. This approach is akin to connecting the end of one arrow to the start of another, forming a chain of vectors. The resulting vector sum, \( \langle 1, 3, 4 \rangle \), is visualized as the direct path from the starting point of the first vector to the ending point of the second vector's tip. This path represents the diagonal of the parallelogram spanned by both vectors.

Geometrically, this resultant vector ('sum vector') can be drawn as starting from the origin and ending at the point (1, 3, 4). This depiction forms a clear representation of vector addition, with the sum vector showing the overall direction and magnitude resulting from the combination.

Component-wise Addition

Component-wise addition is a straightforward method to add vectors in a coordinate system. It involves adding the corresponding components of each vector separately. This technique simplifies complex vector operations by breaking them down into smaller, manageable parts.

For the vectors \( \langle 1, 3, -2 \rangle \) and \( \langle 0, 0, 6 \rangle \), follow these steps:

• Add the x-components: \( 1 + 0 = 1 \)
• Add the y-components: \( 3 + 0 = 3 \)
• Add the z-components: \( -2 + 6 = 4 \)

The completed addition gives us the new vector, \( \langle 1, 3, 4 \rangle \).

This method is advantageous because it ensures that each directional component of the vector is treated independently, allowing for precise control over the final direction and magnitude of the result. Component-wise addition is integral to understanding how multiple forces, directions, or positions can be effectively combined in a mathematical or physical context.