Problem 23 - Precalculus with Calculus Previews

Question

Sketch the vectors \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}\) - \(\mathbf{v}\). $$ \mathbf{u}=5 \mathbf{i}-\mathbf{j}, \mathbf{v}=4 \mathbf{i}-3 \mathbf{j} $$

Step-by-Step Solution

Answer

\( \mathbf{u} + \mathbf{v} = 9\mathbf{i} - 4\mathbf{j} \), \( \mathbf{u} - \mathbf{v} = 1\mathbf{i} + 2\mathbf{j} \).

Step 1: Understand the Given Vectors

The given vectors are \( \mathbf{u} = 5\mathbf{i} - \mathbf{j} \) and \( \mathbf{v} = 4\mathbf{i} - 3\mathbf{j} \). Here, \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors in the x and y directions, respectively. This means that \( \mathbf{u} \) points 5 units in the positive x-direction and 1 unit in the negative y-direction. Similarly, \( \mathbf{v} \) points 4 units in the positive x-direction and 3 units in the negative y-direction.

Step 2: Compute \( \mathbf{u} + \mathbf{v} \)

To find \( \mathbf{u} + \mathbf{v} \), we add the corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \). This gives: \[\begin{align*}\mathbf{u} + \mathbf{v} &= (5\mathbf{i} - \mathbf{j}) + (4\mathbf{i} - 3\mathbf{j}) \&= (5 + 4)\mathbf{i} + (-1 - 3)\mathbf{j} \&= 9\mathbf{i} - 4\mathbf{j}.\end{align*}\]

Step 3: Compute \( \mathbf{u} - \mathbf{v} \)

To find \( \mathbf{u} - \mathbf{v} \), subtract the corresponding components of \( \mathbf{v} \) from \( \mathbf{u} \):\[\begin{align*}\mathbf{u} - \mathbf{v} &= (5\mathbf{i} - \mathbf{j}) - (4\mathbf{i} - 3\mathbf{j}) \&= (5 - 4)\mathbf{i} + (-1 - (-3))\mathbf{j} \&= 1\mathbf{i} + 2\mathbf{j}.\end{align*}\]

Step 4: Sketch the Resultant Vectors

Using the component forms of the resultant vectors, sketch \( \mathbf{u} + \mathbf{v} = 9\mathbf{i} - 4\mathbf{j} \) and \( \mathbf{u} - \mathbf{v} = 1\mathbf{i} + 2\mathbf{j} \). Start each vector from the origin. For \( \mathbf{u} + \mathbf{v} \), move 9 units in the positive x-direction and 4 units in the negative y-direction. For \( \mathbf{u} - \mathbf{v} \), move 1 unit in the positive x-direction and 2 units in the positive y-direction.

Key Concepts

Vector Subtraction

Vector subtraction involves taking two vectors and finding the difference between them. When you subtract vectors, you are essentially reversing the direction of the vector that you are subtracting, and then adding the result to the first vector.

For example, the subtraction of vector \( \mathbf{v} \) from vector \( \mathbf{u} \) is written as \( \mathbf{u} - \mathbf{v} \). To perform this operation, subtract the x-components and the y-components separately.
In our given problem, we have \( \mathbf{u} = 5\mathbf{i} - \mathbf{j} \) and \( \mathbf{v} = 4\mathbf{i} - 3\mathbf{j} \). So the operations for subtraction look like this:

• Subtract x-components: \( 5 - 4 = 1 \)
• Subtract y-components: \( -1 - (-3) = 2 \)

Thus, the resultant vector \( \mathbf{u} - \mathbf{v} \) is \( 1\mathbf{i} + 2\mathbf{j} \), which indicates a motion of 1 unit in the positive x-direction and 2 units in the positive y-direction.

Unit Vectors

Unit vectors are essential in the study of vectors as they provide direction within a coordinate system without contributing to magnitude. The unit vectors in two-dimensional space are usually denoted by \( \mathbf{i} \) and \( \mathbf{j} \).

These vectors represent one unit length in the x-direction and y-direction, respectively. The purpose of using unit vectors is to simplify the representation and calculation of vector components:

• \( \mathbf{i} \) points in the x-direction.
• \( \mathbf{j} \) points in the y-direction.

In the example \( \mathbf{u} = 5\mathbf{i} - \mathbf{j} \), the coefficient 5 before \( \mathbf{i} \) tells us that it extends 5 units along the x-axis. Meanwhile, \( \mathbf{v} = 4\mathbf{i} - 3\mathbf{j} \) extends 4 units along the x-axis and -3 units along the y-axis, indicating movement in the opposite direction.

Resultant Vector

The resultant vector is the vector that results from adding or subtracting two or more vectors together. It's like finding a midpoint or synthesis between two forces acting on the same point.

Consider the resultant of two vectors \( \mathbf{u} \) and \( \mathbf{v} \). When we add these vectors, we add their respective components:

• For \( \mathbf{u}+\mathbf{v} \): add x-components (\( 5 + 4 = 9 \)) and y-components (\( -1 - 3 = -4 \)).

Thus, the resultant vector \( \mathbf{u} + \mathbf{v} \) equals \( 9\mathbf{i} - 4\mathbf{j} \). This resultant tells us the direction and magnitude of the vector formed by composing the effects of both \( \mathbf{u} \) and \( \mathbf{v} \). The resultant vector takes you 9 units east and 4 units south from the origin.
Such calculations help in various fields, including physics and engineering, where combined vector forces are interpreted.