Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Given \(\mathbf{u}=\langle- 2,5\rangle\) and \(\mathbf{v}=\langle 4,3\rangle,\) find each vector. Do not use a calculator. $$-5 \mathbf{v}$$

Step-by-Step Solution

Answer

The vector \(-5 \mathbf{v}\) is \(\langle -20, -15 \rangle\).

Step 1: Understand Scalar Multiplication

Scalar multiplication involves multiplying each component of the vector by the scalar. In this case, we will multiply the vector \( \mathbf{v} = \langle 4, 3 \rangle \) by the scalar \(-5\).

Step 2: Multiply Each Component by the Scalar

Apply the scalar multiplication to each component of \( \mathbf{v} \). This means computing \(-5 \cdot 4\) for the first component and \(-5 \cdot 3\) for the second component.

Step 3: Perform the Multiplications

Calculate the results of the multiplication: \(-5 \cdot 4 = -20\) and \(-5 \cdot 3 = -15\).

Step 4: Write the Resultant Vector

Combine the results from the multiplications into a new vector: \( -5 \mathbf{v} = \langle -20, -15 \rangle \).

Key Concepts

Scalar Multiplication

Scalar multiplication is an essential operation when working with vectors. It's all about scaling a vector by a number, known as a scalar. Consider a vector represented by two components, like \( \langle a, b \rangle \). When you multiply this vector by a scalar, say \( c \), each component of the vector is multiplied by this scalar. So, the resultant vector becomes \( \langle c \cdot a, c \cdot b \rangle \).
For instance, if we have the vector \( \mathbf{v} = \langle 4, 3 \rangle \) and wish to perform the operation \( -5 \cdot \mathbf{v} \), the scalar \( -5 \) is used to scale each component of the vector:

• The first component, 4, is multiplied by -5, giving us -20.
• The second component, 3, is also multiplied by -5, resulting in -15.

Thus, the new vector is \( \langle -20, -15 \rangle \). It's a straightforward method that allows you to stretch or shrink vectors, and even reverse their direction if the scalar is negative.

Vector Components

Vectors are mathematical entities that hold both magnitude and direction. Each vector is made up of components, which are essentially the projections of the vector along the coordinate axes.
The notation \( \langle a, b \rangle \) defines a vector in two-dimensional space, where:

• \( a \) is the component along the x-axis,
• \( b \) is the component along the y-axis.

This notation is crucial because it provides a clear way to handle vectors algebraically. For instance, if you have the vector \( \mathbf{u} = \langle -2, 5 \rangle \), it means that the vector points in the direction defined by these components, with -2 units along the x-axis and 5 units along the y-axis.
Understanding how to interpret these components is key, especially when performing operations such as addition, scalar multiplication, or when you're asked to resolve vectors into their components for further calculations.

Algebraic Operations

Performing algebraic operations on vectors is similar yet distinct from traditional algebra. These operations include addition, subtraction, and scalar multiplication. They are immensely powerful for solving problems involving vectors in physics and engineering.
Here’s a quick overview of common algebraic operations involving vectors:

• Addition: When you add two vectors, like \( \mathbf{u} = \langle a, b \rangle \) and \( \mathbf{v} = \langle c, d \rangle \), you simply add their corresponding components: \( \langle a+c, b+d \rangle \).
• Subtraction: Subtracting vectors follows a similar pattern: \( \langle a, b \rangle - \langle c, d \rangle = \langle a-c, b-d \rangle \).
• Scalar Multiplication: As described earlier, this involves multiplying each component of a vector by a scalar, which changes the vector's magnitude but not its direction unless the scalar is negative.

These operations allow for the manipulation of vectors to achieve desired outcomes in mathematical problems, providing tools to express and solve real-world scenarios effectively.