Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Given \(\mathbf{u}=\langle- 2,5\rangle\) and \(\mathbf{v}=\langle 4,3\rangle,\) find each vector. Do not use a calculator. $$\mathbf{u}-\mathbf{v}$$

Step-by-Step Solution

Answer

The vector difference \( \mathbf{u} - \mathbf{v} \) is \( \langle -6, 2 \rangle \).

Step 1: Write the Expression for the Difference in Component Form

To find the difference of two vectors \( \mathbf{u} \) and \( \mathbf{v} \), write each vector in component form. We have \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \). The expression for their difference is \( \mathbf{u} - \mathbf{v} = \langle -2, 5 \rangle - \langle 4, 3 \rangle \).

Step 2: Subtract Corresponding Components

Subtract the components of vector \( \mathbf{v} \) from vector \( \mathbf{u} \) component-wise. The x-component is \( -2 - 4 \) and the y-component is \( 5 - 3 \).

Step 3: Calculate the Resulting Components

Compute the subtraction for each component: for the x-component, \( -2 - 4 = -6 \) and for the y-component, \( 5 - 3 = 2 \). This gives you the components of the resulting vector.

Step 4: Write Down the Resulting Vector

The difference of the vectors is given by combining the resulting components. Thus, \( \mathbf{u} - \mathbf{v} = \langle -6, 2 \rangle \).

Key Concepts

Component Form

Vectors can be represented in something called the component form. This is a really handy way to see exactly how much a vector stretches in each direction. In two dimensions, a vector like \( \mathbf{u} = \langle -2, 5 \rangle \) tells us that it has:

• a horizontal (or x-direction) component of \(-2\)
• a vertical (or y-direction) component of \(5\)

This means you move 2 units to the left (because of the negative sign) and 5 units up from the origin or your starting point. Expressing vectors in this form makes it simple to perform operations like addition and subtraction by handling each component separately instead of working with the whole vector at once. You just focus on the horizontal parts together and the vertical parts together. This makes calculations much easier and organized.

Resulting Vector

The concept of the resulting vector is quite straightforward once you break it down into parts. When you subtract one vector from another, you're finding a vector that represents this difference. For example, when we calculated \( \mathbf{u} - \mathbf{v} = \langle -2, 5 \rangle - \langle 4, 3 \rangle \), we performed subtraction on each component separately.Here’s how we find the resulting vector:

• The x-component: \(-2 - 4 = -6\)
• The y-component: \(5 - 3 = 2\)

The resulting vector from this operation is \( \langle -6, 2 \rangle \). This new vector tells us that if you move 6 units left and 2 units up from your starting point in space, you'd end up at the tip of the resulting vector. This resulting vector effectively replaces both \( \mathbf{u} \) and \( \mathbf{v} \) to show that difference.

Vector Components

Understanding vector components is important for dissecting a vector's full effect. When we talk about vector components, we refer to the individual parts that together form the vector. These are typically shown as x and y coordinates in two-dimensional space.Consider a vector \( \mathbf{v} = \langle 4, 3 \rangle \):

• The x-component (4) moves horizontally to the right.
• The y-component (3) moves vertically upwards.

When you subtract vectors, you're working directly with these components. Each component tells part of the story, and together they define the vector's direction and magnitude. Working with components allows you to break down complex vector operations into simple arithmetic. It simplifies understanding how individual elements interact, as we did when subtracting the vectors \( \mathbf{u} \) and \( \mathbf{v} \) to get the new vector \( \langle -6, 2 \rangle \).