Problem 25 - Precalculus with Calculus Previews

Question

Sketch the vectors \(2 \mathbf{v}\) and \(-2 \mathbf{v}\). $$ \mathbf{v}=\langle-2,1\rangle $$

Step-by-Step Solution

Answer

\(2 \mathbf{v} = \langle -4, 2 \rangle\) and \(-2 \mathbf{v} = \langle 4, -2 \rangle\).

Step 1: Understand the Vector

The given vector \(\mathbf{v}\) is \(\langle -2, 1 \rangle\). This means the vector points in the direction of -2 units on the x-axis and 1 unit on the y-axis.

Step 2: Calculate the Vector \(2\mathbf{v}\)

To obtain \(2\mathbf{v}\), multiply each component of \(\mathbf{v}\) by 2. This yields \(2\mathbf{v} = 2 \times \langle -2, 1 \rangle = \langle -4, 2 \rangle\).

Step 3: Calculate the Vector \(-2\mathbf{v}\)

Similarly, multiply each component of \(\mathbf{v}\) by -2 to get \(-2\mathbf{v} = -2 \times \langle -2, 1 \rangle = \langle 4, -2 \rangle\).

Step 4: Plot the Original and Scaled Vectors

On a Cartesian coordinate system, plot the original vector \(\mathbf{v} = \langle -2, 1 \rangle\), \(2\mathbf{v} = \langle -4, 2 \rangle\), and \(-2\mathbf{v} = \langle 4, -2 \rangle\). Use arrows to indicate the direction of each vector.

Step 5: Analyze the Scaled Vectors

The vector \(2\mathbf{v}\) is a scaled version of \(\mathbf{v}\) that is twice as long and in the same direction, while \(-2\mathbf{v}\) is also twice as long but in the opposite direction.

Key Concepts

Vector Scaling

Scaling a vector involves multiplying it by a scalar, which is simply a real number. When you scale a vector, you adjust its magnitude, or length, without changing its direction, unless you're multiplying by a negative number (more on this later). In the original exercise, you start with a vector \( \mathbf{v} = \langle -2, 1 \rangle \). To find \( 2\mathbf{v} \), you multiply each component of the vector \( \mathbf{v} \) by 2. Mathematically, it looks like this:

• First component: \( 2 \times (-2) = -4 \)
• Second component: \( 2 \times 1 = 2 \)

This gives you \( 2\mathbf{v} = \langle -4, 2 \rangle \), a vector that's twice as long as the original \( \mathbf{v} \). This concept helps reveal how vectors change in length when scaled, a crucial part of many physics and engineering problems.
It’s important to remember that when you multiply by a scalar:

• The vector becomes longer if the scalar is greater than 1
• Shorter if the scalar is between 0 and 1
• And its length remains unchanged (just reversed in direction if negative).

Negative Vectors

A vector can be considered "negative" when it changes direction due to multiplication by a negative scalar. This leads to a vector that points in the opposite direction of the original. So, when the negative vector is multiplied, each component reverses its sign.
In the exercise, the original vector is \( \mathbf{v} = \langle -2, 1 \rangle \). To determine \( -2\mathbf{v} \), you multiply each component by \(-2\).

• First component: \( -2 \times (-2) = 4 \)
• Second component: \( -2 \times 1 = -2 \)

This results in the negative scaled vector \(-2\mathbf{v} = \langle 4, -2 \rangle\). Notice how the signs in the vector components change, thereby reversing the direction.
The length of this negative vector is the same as that of \(2\mathbf{v}\), but now it points in the opposite direction. Recognizing this allows you to understand directional shifts in vectors, which has practical applications in navigation and motion planning.

Coordinate System Plotting

The Cartesian coordinate system is designed to plot and visualize vectors clearly. Understanding how to plot vectors helps in visualizing their direction and magnitude, making it easier to interpret data in physics and engineering contexts.
For vectors like \( \mathbf{v} = \langle -2, 1 \rangle \), \( 2\mathbf{v} = \langle -4, 2 \rangle \), and \(-2\mathbf{v} = \langle 4, -2 \rangle \), you start by plotting the origin point. The tail of each vector generally begins at the origin (0,0), and the head points to the coordinates represented by the vector components, e.g., \(-4, 2\).

• Mark the head of a vector by counting units on the x-axis and y-axis.
• Draw an arrow starting from the origin to help visualize direction and length.
• Compare vector lengths to see how scaling affects the plot visually.

This plotting method will strongly aid in your ability to solve real-world problems using vector mathematics, such as calculating forces or velocities in physics.