Problem 10 - Essential Calculus Early Transcendentals

Question

Find the sum of the given vectors and illustrate geometrically. $$\langle 3,-1\rangle, \quad\langle- 1,5\rangle$$

Step-by-Step Solution

Answer

The sum of the vectors is \( \langle 2, 4 \rangle \).

Step 1: Identify the Vectors

The vectors given in the problem are \( \langle 3, -1 \rangle \) and \( \langle -1, 5 \rangle \). Each vector is represented by its components along the x-axis and y-axis.

Step 2: Perform Vector Addition

To find the sum of these vectors, add their corresponding components separately. This means adding the x-components together: \( 3 + (-1) = 2 \), and the y-components: \( -1 + 5 = 4 \). The resultant vector is therefore \( \langle 2, 4 \rangle \).

Step 3: Interpret the Result

The resulting vector \( \langle 2, 4 \rangle \) tells us the combined effect of the original vectors when they are added. This vector starts from the origin and points to the position (2, 4) on the coordinate plane.

Step 4: Illustrate Vector Addition Geometrically

In a geometric representation, draw the first vector \( \langle 3, -1 \rangle \), starting from the origin. From the endpoint of this vector, draw the second vector \( \langle -1, 5 \rangle \). The vector sum, \( \langle 2, 4 \rangle\), can be obtained by drawing a vector from the origin to the endpoint of the second vector. This forms a 'triangle' or parallelogram, where the diagonal represents the resultant vector.

Key Concepts

Understanding Vector Components

Vectors are mathematical entities characterized by both magnitude and direction. To effectively solve problems involving vectors, it's crucial to understand their components. Every vector can be broken down into two parts: the horizontal (x-axis) component and the vertical (y-axis) component. For instance, in the vector \( \langle 3, -1 \rangle \), the x-component is 3, and the y-component is -1. Similarly, in \( \langle -1, 5 \rangle \), the x-component is -1 and the y-component is 5.

These components represent how far the vector moves on each axis. Think of them as directions: the x-component shows side-to-side movement, while the y-component indicates up-and-down movement. When solving problems, this breakdown simplifies calculations and provides the groundwork for adding vectors or visualizing them geometrically.

• Horizontal component determines the "side" movement.
• Vertical component determines the "up" movement.

Recognizing these components helps in computing the overall effect of multiple vectors or visualizing them in diagrams on a coordinate plane.

Effectively Finding the Resultant Vector

When multiple vectors act simultaneously, it is often necessary to find a single vector that has the same effect as those acting in parallel. This single vector is known as the resultant vector. Calculating the resultant vector involves summing up the individual vector components of the given vectors.

Using the example of vectors \( \langle 3, -1 \rangle \) and \( \langle -1, 5 \rangle \), we add the respective components: the x-components (3 and -1) add up to 2, and the y-components (-1 and 5) add up to 4. Thus, the resultant vector becomes \( \langle 2, 4 \rangle \). This new vector maintains the essence of the overall direction and magnitude influenced by both initial vectors.

• Add corresponding components to find the resultant vector.
• The resultant vector simplifies the problem of managing individual vectors.

Understanding the resultant vector is crucial in physics and engineering, where combining forces and motions is a common task.

The Geometric Representation of Vectors

Visually representing vectors is a powerful way to comprehend their interaction and sum. The geometric representation forms an intuitive understanding that complements the arithmetic approach.

To visualize the addition of vectors such as \( \langle 3, -1 \rangle \) and \( \langle -1, 5 \rangle \), begin by drawing the first vector from the origin to the point (3, -1) on a coordinate plane. Next, draw the second vector starting where the first vector ends—moving to the point determined by its components. This sequence forms a path known as the vector chain.

The resultant vector is drawn from the starting point (origin) directly to the endpoint of the vector chain, this line represents the overall effect of combining the vectors. Such a representation often takes the shape of a geometric figure, like a triangle or parallelogram, where the diagonal is the resultant vector.

• Start drawing from the origin along the x and y components.
• Link the endpoint of one vector to the beginning of another.
• The resultant is the direct line from the origin to the vector chain's end.

Using a geometric approach not only aids in verifying arithmetic calculations but also enriches one’s conceptual grasp on how vectors interact spatially.