Problem 57 - Precalculus with Limits

Question

In Exercises 57-62, find the component form of \(\small{\mathbf{v}}\) and sketchthe specified vector operations geometrically, where \(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\), and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \frac{3}{2} \mathbf{u}}\)

Step-by-Step Solution

Answer

\(\small{\mathbf{v} = 3\mathbf{i} - \frac{3}{2}\mathbf{j}}\)

Step 1: Identify Given Vectors

We have two vectors:\(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\) and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\)

Step 2: Write the Vector Equation

The vector \(\small{\mathbf{v}}\) is defined as: \(\small{\mathbf{v} = \frac{3}{2} \mathbf{u}}\)

Step 3: Substitute \(\small{\mathbf{u}}\) into the Equation

Substitute \(\small{\mathbf{u}}\) into the equation, we get \(\small{\mathbf{v} = \frac{3}{2} (2\mathbf{i} - \mathbf{j})}\)

Step 4: Calculate the Components

Directly multiply the scalar into the vector, \(\small{\mathbf{u}}\), we get \(\small{\mathbf{v} = 3\mathbf{i} - \frac{3}{2}\mathbf{j}}\)

Key Concepts

Understanding Component Form

In vector mathematics, the component form of a vector breaks it into its individual parts or dimensions. This makes it easier to analyze and compute vector operations. For instance, a vector in component form shows how much it extends along the horizontal (x-axis) and vertical (y-axis).

For example, the vector \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) in the original problem is expressed in component form. Here, the 2 and -1 are the horizontal and vertical components, respectively. This can be visualized as \((2, -1)\).

The familiarity with the component form allows us to perform operations like addition, subtraction, or multiplication more straightforwardly. When vectors are broken down into components, it becomes much simpler to handle each part separately and perform precise calculations.

Exploring Scalar Multiplication

Scalar multiplication is a fundamental operation in vector mathematics where a vector is multiplied by a scalar (a constant number). This operation affects both the magnitude and direction of the vector, based upon the scalar's value.

When you multiply a vector \( \mathbf{u} \) by a scalar, each component of \( \mathbf{u} \) is multiplied by that scalar. For instance, if \( \mathbf{u} = (2, -1) \) and we want to compute \( \frac{3}{2}\mathbf{u} \), we multiply each component by \( \frac{3}{2} \).

This results in new components: \( \frac{3}{2} \times 2 = 3 \) and \( \frac{3}{2} \times -1 = -\frac{3}{2} \). Therefore, the scalar multiplication of \( \mathbf{u} \) yields a new vector \( \mathbf{v} = (3, -\frac{3}{2}) \).

• The direction remains the same if the scalar is positive, but reverses if negative.
• The magnitude scales up or down based on whether the scalar is greater than or less than one.

Performing Vector Operations

Vector operations are the calculations we perform using vectors. These include addition, subtraction, and scalar multiplication. They allow us to understand the effects of different forces or movements described by vectors in physics and engineering contexts.

Let's consider the vector operation from the original problem where \( \mathbf{v} = \frac{3}{2} \mathbf{u} \). This involves scalar multiplication as we've seen before. Here’s a brief rundown of typical vector operations:

• Addition: Add corresponding components from two vectors. For example, \( (x_1, y_1) + (x_2, y_2) = (x_1+x_2, y_1+y_2) \).
• Subtraction: Subtract corresponding components. E.g., \( (x_1, y_1) - (x_2, y_2) = (x_1-x_2, y_1-y_2) \).
• Scalar Multiplication: As discussed, multiply each component by the scalar.

Each of these operations can be visualized geometrically which helps in understanding how vectors interact spatially. Graphic representation makes the outcome of vector combinations intuitive to grasp, offering insight into spatial dimensions and movements.