Problem 58 - 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 \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \frac{3}{4} \mathbf{w}}\)

Step-by-Step Solution

Answer

The component form of vector v is 0.75i + 1.5j.

Step 1 Read the Problem and understand the provided vectors

The problem provides two vectors, named u and w. The vector u is given as 2i - j and the vector w is given as i + 2j. We are interested in vector v, which is 3/4 of vector w. Here, i and j are the component form notation of vectors.

Step 2 Express Vector w in component form

Start off by writing down the individual i and j components of the given vector w. This becomes essential to visualize and understand the vector. Thus, vector w is (1,2) because i, the horizontal component, is 1 and j, the vertical component, is 2.

Step 3 Calculate Vector v

After knowing the component form of vector w, we need to figure out vector v, which is 3/4th of vector w. Multiply each component of vector w by 3/4 to get the components of vector v. Hence, v = (3/4 * 1, 3/4 * 2) = (0.75,1.5).

Step 4 Write down the component form and sketch the vector

Finally, the component form of the vector v is written as v = 0.75i + 1.5j in terms of i and j components. This completed the first part of the problem. The second part of the problem requires sketching the vectors, which is done on a graph considering i and j components as the x and y coordinates respectively.

Key Concepts

Component Form

When working with vectors, it's crucial to understand their component form. Vectors like \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} + 2\mathbf{j} \) are expressed using unit vectors \( \mathbf{i} \) and \( \mathbf{j} \). These represent the horizontal and vertical components respectively.

To convert these vectors into component form, we extract the coefficients of \( \mathbf{i} \) and \( \mathbf{j} \). For example:

• \( \mathbf{u} = (2, -1) \)
• \( \mathbf{w} = (1, 2) \)

In this format, the first number is the horizontal component, while the second is the vertical component. It helps break down complex vector operations into manageable steps.

Vector Addition

Combining vectors is straightforward once they are in component form. Let's say you want to add \( \mathbf{u} = (2, -1) \) and another vector \( \mathbf{w} = (1, 2) \). You simply add corresponding components together:

• Horizontal: \( 2 + 1 = 3 \)
• Vertical: \( -1 + 2 = 1 \)

This results in the new vector \( \mathbf{u} + \mathbf{w} = (3, 1) \).

Vector addition visualizes as placing one vector at the tip of the other, forming a new diagonal vector that represents the resultant force or movement.

Scalar Multiplication

Scalar multiplication involves scaling a vector by a single number. Suppose we have \( \mathbf{w} = (1, 2) \) and we want \( \mathbf{v} = \frac{3}{4}\mathbf{w} \).

Multiply each component by \( \frac{3}{4} \):

• Horizontal: \( \frac{3}{4} \times 1 = 0.75 \)
• Vertical: \( \frac{3}{4} \times 2 = 1.5 \)

Therefore, \( \mathbf{v} = (0.75, 1.5) \).

This scaling changes the length of the vector without altering its direction. It’s equivalent to stretching or shrinking the vector based on the scalar value.

Sketching Vectors

Sketching vectors offers a visual perspective, making it easier to interpret and analyze their properties. On a graph, each vector can be represented by starting at the origin \((0,0)\) and extending to the point defined by its components.

For instance, sketching \( \mathbf{v} = (0.75, 1.5) \):

• Start at the origin \((0,0)\).
• Move 0.75 units horizontally and 1.5 units vertically.

The endpoint \((0.75, 1.5)\) is where the vector's arrowhead lands.

Using the geometric representation simplifies understanding, especially for operations like addition or subtraction, by showing how vectors combine or differ in a spatial sense.