Problem 28 - Precalculus with Calculus Previews

Question

Sketch the vectors \(2 \mathbf{v}\) and \(-2 \mathbf{v}\). $$ \mathbf{v}=-\frac{1}{2} \mathbf{i}+\frac{3}{2} \mathbf{j} $$

Step-by-Step Solution

Answer

Scale components of \(\mathbf{v}\) by 2 and -2 for vectors \(2\mathbf{v}, -2\mathbf{v}\), then sketch.

Step 1: Understand Vector Components

The vector \( \mathbf{v} \) is given as \( \mathbf{v} = -\frac{1}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \). This means it can be decomposed into its components: -\(\frac{1}{2} \) in the x-direction (\( \mathbf{i} \)) and \(\frac{3}{2} \) in the y-direction (\( \mathbf{j} \)). We will be scaling these components by 2 and by -2 to get the vectors \( 2\mathbf{v} \) and \(-2\mathbf{v} \).

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

To find \(2 \mathbf{v} \), multiply each component of \( \mathbf{v} \) by 2: \[2 \mathbf{v} = 2 \left( -\frac{1}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \right) = -\mathbf{i} + 3 \mathbf{j}.\] Thus, the vector \( 2\mathbf{v} \) has components (-1, 3).

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

To find \(-2 \mathbf{v} \), multiply each component of \( \mathbf{v} \) by -2: \[-2 \mathbf{v} = -2 \left( -\frac{1}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \right) = \mathbf{i} - 3 \mathbf{j}.\] Thus, the vector \( -2\mathbf{v} \) has components (1, -3).

Step 4: Sketch the Vectors

Draw the original vector \( \mathbf{v} \) from the origin with components \((-\frac{1}{2}, \frac{3}{2})\). For \(2 \mathbf{v} \), draw a vector from the origin to the point \((-1, 3)\). For \(-2 \mathbf{v} \), draw a vector from the origin to the point \((1, -3)\). Ensure that \(2 \mathbf{v} \) is in the same direction as \( \mathbf{v} \) but longer, and \(-2 \mathbf{v} \) is in the opposite direction.

Key Concepts

Vector Components

A vector is a mathematical entity that has both magnitude and direction. The components of a vector are values that represent its projection along the coordinate axes. In this case, our vector \( \mathbf{v} \) is given in terms of its components along the \(x\)-axis and the \(y\)-axis. These axes are represented by \( \mathbf{i} \) and \( \mathbf{j} \) respectively.
For the vector \( \mathbf{v} = -\frac{1}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \), it has two components:

• \(-\frac{1}{2}\) along the \(x\)-axis (\( \mathbf{i} \))
• \(\frac{3}{2}\) along the \(y\)-axis (\( \mathbf{j} \))

These components allow us to understand and manipulate the vector, such as scaling it to find \(2 \mathbf{v}\) or \(-2 \mathbf{v}\) by multiplying each component by a scalar factor.

Vector Notation

Vectors are often expressed in a form that clearly shows their components. This is known as vector notation, which simplifies the understanding and communication of vector information. For a vector \( \mathbf{v} \), vector notation uses bold letters, typically showing how it can be broken down into its directional components.
The mathematical representation often includes standard unit vectors:

• \( \mathbf{i} \) denotes the unit vector in the direction of the \(x\)-axis
• \( \mathbf{j} \) denotes the unit vector in the direction of the \(y\)-axis

So, the vector notation for \( \mathbf{v} \) would be \(-\frac{1}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \), meaning the vector moves half a unit in the negative \(x\)-direction and one and a half units in the positive \(y\)-direction.

Vector Direction

The direction of a vector is an essential aspect as it indicates where the vector points. In our example, \( \mathbf{v} = -\frac{1}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \), its direction is derived from its components. The \(-\frac{1}{2} \mathbf{i} \) component suggests a movement to the left along the \(x\)-axis, while the \(\frac{3}{2} \mathbf{j} \) component means a move upwards along the \(y\)-axis.
To better understand vector multiplication's effect on direction, note that:

• A positive scalar keeps the direction the same.
• A negative scalar reverses the direction.

Thus, multiplying \( \mathbf{v} \) by 2 results in a vector \( 2\mathbf{v} \) with the same direction, while multiplying by -2 alters the direction, making \( -2\mathbf{v} \) point in the opposite direction.

Sketching Vectors

Sketching vectors is a way to visualize their magnitude and direction on a coordinate plane. The original vector, \( \mathbf{v} = -\frac{1}{2} \mathbf{i} + \frac{3}{2} \mathbf{j} \), can be plotted by starting at the origin, then moving \(-\frac{1}{2}\) units left (on the \(x\)-axis) and \(\frac{3}{2}\) units up (on the \(y\)-axis).
When sketching \(2 \mathbf{v}\) and \(-2 \mathbf{v}\), follow these steps:

• For \(2 \mathbf{v}\): Double the move indicated by \( \mathbf{v} \), leading to \(-1\) left, and \(3\) up, ending at \((-1, 3)\).
• For \(-2 \mathbf{v}\): Reverse and double the move of \( \mathbf{v} \), reaching \(1\) right, and \(-3\) down, ending at \((1, -3)\).

This visualization helps understand how vector components scale with multiplication and how they express direction effectively.