Problem 30 - 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 is \( \langle 20, 15 \rangle \).

Step 1: Understand the Problem

The problem asks for the vector resulting from multiplying the given vector \( \mathbf{v} = \langle 4, 3 \rangle \) by the scalar 5.

Step 2: Recall Scalar Multiplication

Scalar multiplication involves multiplying each component of the vector by the scalar. For a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) and a scalar \( c \), the operation \( c \mathbf{a} \) results in the vector \( \langle c \cdot a_1, c \cdot a_2 \rangle \).

Step 3: Apply Scalar Multiplication

Apply the scalar multiplication to the vector \( \mathbf{v} = \langle 4, 3 \rangle \) with the scalar 5. Calculate: \( 5 \times 4 = 20 \) and \( 5 \times 3 = 15 \).

Step 4: Write the Result

The resultant vector of the multiplication is \( \langle 20, 15 \rangle \).

Key Concepts

Scalar Multiplication

Scalar multiplication is a foundational concept in vector operations. When you perform scalar multiplication, you multiply each component of a vector by a real number, known as the scalar. This operation effectively scales the magnitude of the vector without changing its direction. For example, if you have a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \) and you multiply it by a scalar \( c \), the resultant vector becomes \( \langle c \cdot a_1, c \cdot a_2 \rangle \). In this operation, every component of the vector grows proportionally to the scalar.

**Steps to Calculate Scalar Multiplication:**

• Identify the vector components and the scalar.
• Multiply each component of the vector by the scalar separately.
• Combine these new values into the resultant vector.

This simple arithmetic operation allows you to easily find a new vector that lies on the same line as the original but differs in length.

Resultant Vector

A resultant vector is the vector obtained after performing an operation, such as scalar multiplication or addition, on one or more vectors. It tells us the end point of the vector operation journey.
For instance, when you multiply a vector by a scalar, like in our given problem, you create a resultant vector that is a scaled version of the original vector. Here, multiplying \( \mathbf{v} = \langle 4, 3 \rangle \) by the scalar 5 yields the resultant vector \( \langle 20, 15 \rangle \).

The direction of the resultant vector remains the same as the original vector's unless the scalar becomes negative, which would reverse the direction. Importantly, calculating the resultant vector doesn't just mean finding an endpoint; it also has practical implications in physics and engineering, where directional movements and forces are represented as vectors. This helps in comprehending dimensions and effects in real-world applications.

Vector Operations

Vector operations encompass a variety of manipulations you can perform on vectors, such as addition, subtraction, and scalar multiplication. Each type of operation alters vectors in distinct ways to achieve different results.

**Types of Vector Operations:**

• Addition: To add two vectors, simply add their respective components. For example, \( \mathbf{u} = \langle x_1, y_1 \rangle \) and \( \mathbf{v} = \langle x_2, y_2 \rangle \) can be added to get \( \langle x_1 + x_2, y_1 + y_2 \rangle \).
• Subtraction: Similar to addition, but you subtract the components instead. The result, or difference vector, is \( \langle x_1 - x_2, y_1 - y_2 \rangle \).
• Scalar Multiplication: As described, this involves multiplying every component of a vector by a scalar, resulting in a change in the vector's magnitude.

Understanding vector operations is crucial as they form the backbone of more complex mathematical modeling and problem solving. Each operation can be visualized geometrically, providing an intuitive understanding of mathematical concepts and aiding in fields like engineering, physics, and computer graphics.