Problem 21 - Precalculus with Calculus Previews

Question

Sketch the vectors \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}\) - \(\mathbf{v}\). $$ \mathbf{u}=2 \mathbf{i}+3 \mathbf{j}, \mathbf{v}=-\mathbf{i}+2 \mathbf{j} $$

Step-by-Step Solution

Answer

\(\mathbf{u} + \mathbf{v} = \mathbf{i} + 5 \mathbf{j}\) and \(\mathbf{u} - \mathbf{v} = 3\mathbf{i} + \mathbf{j}\).

Step 1: Understand the Given Vectors

Two vectors are given: \(\mathbf{u} = 2 \mathbf{i} + 3 \mathbf{j}\) and \(\mathbf{v} = -\mathbf{i} + 2 \mathbf{j}\). These vectors are in component form where \(\mathbf{i}\) and \(\mathbf{j}\) represent the unit vectors along the x-axis and y-axis respectively.

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

To find \(\mathbf{u} + \mathbf{v}\), add the corresponding components of \(\mathbf{u}\) and \(\mathbf{v}\): \((2 \mathbf{i} + 3 \mathbf{j}) + (-\mathbf{i} + 2 \mathbf{j}) = (2 + (-1)) \mathbf{i} + (3 + 2) \mathbf{j} = 1 \mathbf{i} + 5 \mathbf{j}\). So, \(\mathbf{u} + \mathbf{v} = \mathbf{i} + 5 \mathbf{j}\).

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

To find \(\mathbf{u} - \mathbf{v}\), subtract the components of \(\mathbf{v}\) from those of \(\mathbf{u}\): \((2 \mathbf{i} + 3 \mathbf{j}) - (-\mathbf{i} + 2 \mathbf{j}) = (2 - (-1)) \mathbf{i} + (3 - 2) \mathbf{j} = 3 \mathbf{i} + 1 \mathbf{j}\). So, \(\mathbf{u} - \mathbf{v} = 3 \mathbf{i} + \mathbf{j}\).

Step 4: Sketch the Vectors

To sketch the vectors, use the component form of each vector as coordinates. For \(\mathbf{u} + \mathbf{v}\), start at the origin and draw the vector to the point (1, 5). For \(\mathbf{u} - \mathbf{v}\), draw the vector to the point (3, 1). Ensure each vector is accurately placed on a coordinate grid.

Key Concepts

Component Form of Vectors

When we talk about vectors in the context of physics or mathematics, they are often expressed in their component form. This means breaking down a vector into its fundamental parts along individual axes, typically using the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \). These unit vectors are aligned with the coordinate grid axes, specifically:

• \( \mathbf{i} \) aligns with the x-axis.
• \( \mathbf{j} \) aligns with the y-axis.

In the problem, we have two vectors, \( \mathbf{u} = 2 \mathbf{i} + 3 \mathbf{j} \) and \( \mathbf{v} = -\mathbf{i} + 2 \mathbf{j} \). Here, the numbers before \( \mathbf{i} \) and \( \mathbf{j} \) specify how far the vector moves along the x and y axes, respectively. This form is particularly useful for operations like addition and subtraction, as you can add or subtract corresponding components directly.

Unit Vectors

Unit vectors play a crucial role in vector mathematics. They are essentially vectors with a magnitude of one and are used to define directions along the different axes. In a 2D space, unit vectors are \( \mathbf{i} \) and \( \mathbf{j} \):

• \( \mathbf{i} \) indicates the direction of the positive x-axis and has components \((1, 0)\).
• \( \mathbf{j} \) indicates the direction of the positive y-axis and has components \((0, 1)\).

Using unit vectors, any vector can be represented as a combination of \( \mathbf{i} \) and \( \mathbf{j} \). For example, the vector \( \mathbf{u} = 2\mathbf{i} + 3\mathbf{j} \) means it moves 2 units along the x-axis and 3 units along the y-axis. This makes it easier to visualize and calculate with vectors on a coordinate grid.

Coordinate Grid

A coordinate grid is essential in visualizing vectors and performing vector operations like addition and subtraction. The grid is made up of horizontal and vertical lines that intersect to form squares, creating reference points for plotting vectors.Vectors are typically sketched starting at the origin, the point (0,0), and extending according to their component form. So, for a vector given as \( 2\mathbf{i} + 3\mathbf{j} \), it starts at the origin and points towards (2,3).The calculated vectors from the exercise, \( \mathbf{u} + \mathbf{v} = \mathbf{i} + 5\mathbf{j} \) and \( \mathbf{u} - \mathbf{v} = 3\mathbf{i} + \mathbf{j} \), can be plotted using this method:

• For \( \mathbf{u} + \mathbf{v} \), the vector points towards (1,5).
• For \( \mathbf{u} - \mathbf{v} \), it points towards (3,1).

Using a coordinate grid allows for a precise graphical representation of vectors, which is extremely helpful for understanding their direction and magnitude visually.