Problem 22 - Precalculus with Calculus Previews

Question

Sketch the vectors \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}\) - \(\mathbf{v}\). $$ \mathbf{u}=-4 \mathbf{i}+\mathbf{j}, \mathbf{v}=2 \mathbf{i}+2 \mathbf{j} $$

Step-by-Step Solution

Answer

\(\mathbf{u} + \mathbf{v} = -2\mathbf{i} + 3\mathbf{j}\), \(\mathbf{u} - \mathbf{v} = -6\mathbf{i} - \mathbf{j}\).

Step 1: Calculate \(\mathbf{u} + \mathbf{v}\)

To add vectors \(\mathbf{u} = -4\mathbf{i} + \mathbf{j}\) and \(\mathbf{v} = 2\mathbf{i} + 2\mathbf{j}\), you simply add the corresponding components.- \(\mathbf{i}\) components: \(-4 + 2 = -2\)- \(\mathbf{j}\) components: \(1 + 2 = 3\)Thus, \(\mathbf{u} + \mathbf{v} = -2\mathbf{i} + 3\mathbf{j}\).

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

To subtract vectors, subtract the corresponding components of \(\mathbf{v}\) from \(\mathbf{u}\):- \(\mathbf{i}\) components: \(-4 - 2 = -6\)- \(\mathbf{j}\) components: \(1 - 2 = -1\)Thus, \(\mathbf{u} - \mathbf{v} = -6\mathbf{i} - \mathbf{j}\).

Step 3: Sketch \(\mathbf{u} + \mathbf{v}\)

Start at the origin (0,0). To draw \(\mathbf{u} + \mathbf{v} = -2\mathbf{i} + 3\mathbf{j}\), move 2 units to the left along the \(x\)-axis and 3 units up along the \(y\)-axis. Mark the endpoint and draw an arrow from the origin to this point.

Step 4: Sketch \(\mathbf{u} - \mathbf{v}\)

Start at the origin (0,0). For \(\mathbf{u} - \mathbf{v} = -6\mathbf{i} - \mathbf{j}\), move 6 units to the left along the \(x\)-axis and 1 unit down along the \(y\)-axis. Mark the endpoint and draw an arrow from the origin to this point.

Key Concepts

Component Form of Vectors

Understanding the component form of vectors is crucial when dealing with vector calculations. Vectors are often expressed using their components along the axes of a coordinate system, commonly the \((x, y)\) axes. Each component represents a vector's influence in a particular direction.
For example, the vector \(\mathbf{u} = -4\mathbf{i} + \mathbf{j}\) is expressed in terms of two components:

• The \(\mathbf{i}\) component, which is the horizontal influence, equals \(-4\).
• The \(\mathbf{j}\) component, which is the vertical influence, equals \(1\).

This means that vector \(\mathbf{u}\) points 4 units to the left and 1 unit upwards. Similarly, for vector \(\mathbf{v} = 2\mathbf{i} + 2\mathbf{j}\), the vector points 2 units to the right and 2 units upwards.
Expressing vectors in component form is especially helpful for performing vector addition and subtraction because it allows us to work with numerical values for each direction separately.

Vector Sketching

Sketching vectors is a visual way of understanding vector magnitude and direction. It helps to see how vectors add or subtract to form new vectors which is a foundational concept in physics and engineering.
To sketch a vector such as \(\mathbf{u} + \mathbf{v} = -2\mathbf{i} + 3\mathbf{j}\), you start at the origin, which is the point (0,0).

• Move 2 units to the left because of the \(\mathbf{i}\) component \(-2\).
• Move 3 units up because of the \(\mathbf{j}\) component \(3\).

After achieving the correct position, draw an arrow from the origin to this point. This arrow is the vector.
For subtraction, you'll follow a similar process. For instance, with \(\mathbf{u} - \mathbf{v} = -6\mathbf{i} - \mathbf{j}\), you’d:

• Move 6 units to the left from the origin.
• Then move 1 unit down.

Again, sketch the arrow from the origin to this final point.
These sketching techniques help not only in understanding individual vectors but also in visualizing how they combine to form resultant vectors in operations.

Basic Vector Operations

Vector addition and subtraction are fundamental operations when working with vectors in physics or engineering. These operations allow us to combine vectors to find resultant vectors or to understand how objects, forces, or velocities relate to one another.
In vector addition, you add the corresponding components from each vector. For example, to find \(\mathbf{u} + \mathbf{v}\) where \(\mathbf{u} = -4\mathbf{i} + \mathbf{j}\) and \(\mathbf{v} = 2\mathbf{i} + 2\mathbf{j}\), you calculate:

• Horizontal, or \(\mathbf{i}\), component: \(-4 + 2 = -2\)
• Vertical, or \(\mathbf{j}\), component: \(1 + 2 = 3\)

This results in a new vector \(\mathbf{u} + \mathbf{v} = -2\mathbf{i} + 3\mathbf{j}\).
Similarly, in vector subtraction, the components of one vector are subtracted from the other. So, \(\mathbf{u} - \mathbf{v}\) results in:

• Horizontal component: \(-4 - 2 = -6\)
• Vertical component: \(1 - 2 = -1\)

resulting in \(\mathbf{u} - \mathbf{v} = -6\mathbf{i} - \mathbf{j}\).
These operations are intuitive when vectors are expressed in component form and are instrumental in understanding how vectors interact in both two-dimensional and three-dimensional space.