Problem 59 - 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 \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \mathbf{u} + 2\mathbf{w}}\)

Step-by-Step Solution

Answer

The component form of the vector \(\mathbf{v}\) is 4\(\mathbf{i}\) + 3\(\mathbf{j}\).

Step 1: Identify the Vectors

The vectors are identified from the problem statement as: \(\mathbf{u} = 2\mathbf{i} - \mathbf{j}\) and \(\mathbf{w} = \mathbf{i} + 2\mathbf{j}\)

Step 2: Compute 2\(\mathbf{w}\)

Multiply the vector \(\mathbf{w}\) by 2 to yield 2\(\mathbf{w}\) = 2(\(\mathbf{i} + 2\mathbf{j}\)) = 2\(\mathbf{i}\) + 4\(\mathbf{j}\)

Step 3: Find the Component Form of \(\mathbf{v}\)

Then, calculate the component form of \(\mathbf{v}\) by adding \(\mathbf{u}\) and 2\(\mathbf{w}\): \(\mathbf{v} = \mathbf{u} + 2\mathbf{w} = (2\mathbf{i} - \mathbf{j}) + (2\mathbf{i} + 4\mathbf{j}) = 4\mathbf{i} + 3\mathbf{j}\)

Step 4: Draw the Vectors

To visualize this vector, we plot each of the vectors \(\mathbf{u}\), \(\mathbf{w}\) and \(\mathbf{v}\) in the same graph, based on their component forms, keeping in mind that a positive \(\mathbf{i}\)-coordinate moves us to the right on the x-axis and a positive \(\mathbf{j}\)-coordinate moves us upward on the y-axis.

Key Concepts

Component Form

The component form of a vector is a simple way to express a vector using the unit vectors along the coordinate axes. A vector, like \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \), is broken down into parts that describe the movement along the x-axis (\( \mathbf{i} \)) and y-axis (\( \mathbf{j} \)).
Each coefficient of these unit vectors tells us how far and in which direction to move along each axis.
For example:

• The \( 2\mathbf{i} \) indicates moving 2 units to the right on the x-axis.
• The \( -\mathbf{j} \) means moving 1 unit downwards on the y-axis.

Ultimately, this way of writing vectors makes it easy to perform vector operations like addition and subtraction. You just add or subtract each component separately.

Vector Addition

Vector addition is a fundamental operation where two or more vectors are combined. In our example, we are required to find the vector \( \mathbf{v} = \mathbf{u} + 2\mathbf{w} \). The first step was to multiply the vector \( \mathbf{w} \) by 2, giving us \( 2\mathbf{i} + 4\mathbf{j} \).
Next, we added this new vector, \( 2\mathbf{w} \), to the vector \( \mathbf{u} \) by combining the \( \mathbf{i} \) and \( \mathbf{j} \) components separately:

• For the x-component: \( 2\mathbf{i} + 2\mathbf{i} = 4\mathbf{i} \).
• For the y-component: \( -\mathbf{j} + 4\mathbf{j} = 3\mathbf{j} \).

Thus, the resulting vector \( \mathbf{v} \) has the component form \( 4\mathbf{i} + 3\mathbf{j} \). Vector addition respects the associative and commutative properties, making it straightforward and predictable.

Graphical Representation

Graphical representation of vectors provides a visual insight into how they interact. Once we have determined the component form of our vectors, we can plot them on a graph.
The vector \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) starts at the origin and ends at the point (2, -1), moving 2 units to the right and 1 unit down.
For the vector \( \mathbf{w} = \mathbf{i} + 2\mathbf{j} \), it starts at (0,0) and ends at (1, 2), moving to the right 1 unit and up 2 units.

• Vector \( 2\mathbf{w} \) would thusly extend from (0,0) to (2,4).
• The resulting vector \( \mathbf{v} \) plots from the origin to (4,3), illustrating a total rightward movement of 4 units and an upward movement of 3 units.

Graphically stacking these vectors helps in understanding their resultant vector \( \mathbf{v} \), showing paths through space and the outcome of their addition, mapped with clarity on the coordinate plane.