Problem 4 - Precalculus : Building Concepts and Connections

Question

find \(\mathbf{v} \cdot \mathbf{w}\) $$\mathbf{v}=\left\langle\frac{3}{2},-1\right\rangle, \mathbf{w}=\langle 4,0\rangle$$

Step-by-Step Solution

Answer

The dot product of vectors \(\mathbf{v}\) and \(\mathbf{w}\) is 6.

Step 1: Identification of the vectors

The vectors given are \(\mathbf{v}=\left\langle\frac{3}{2},-1\right\rangle\) and \(\mathbf{w}=\langle 4,0\rangle\)

Step 2: Dot Product Calculation - Part 1

The dot product of \(\mathbf{v}\) and \(\mathbf{w}\) is calculated as the sum of the products of their corresponding entries. First, multiply the first entry of \(\mathbf{v}\) with the first entry of \(\mathbf{w}\) = (\(\frac{3}{2}\)) * 4 = 6.

Step 3: Dot Product Calculation - Part 2

Next, multiply the second entry of \(\mathbf{v}\) with the second entry of \(\mathbf{w}\) = -1 * 0 = 0.

Step 4: Final Calculation

Now, add up the results gotten from Part 1 and Part 2 = 6 + 0 = 6

Key Concepts

Vectors

Vectors are essential components in mathematics, especially when dealing with quantities that require both magnitude and direction. In this exercise, you encounter vector representations with each vector having two components.
The notation \( \mathbf{v} = \left\langle \frac{3}{2}, -1 \right\rangle \) and \( \mathbf{w} = \langle 4, 0 \rangle \) is commonly used to show vectors in two-dimensional space.
Each vector has a horizontal component (first number) and a vertical component (second number).

• Visualizing Vectors: Imagine arrows on a coordinate grid where each vector's tail is at the origin, and its tip points to the coordinates given, such as \( \frac{3}{2}, -1 \).
• Magnitude and Direction: Magnitude represents the length while direction is the angle or way it points. For a vector \( \langle x, y \rangle \), its magnitude is calculated as \( \sqrt{x^2 + y^2} \).

Understanding vectors in this manner simplifies complex topics in physics and engineering where such representations naturally apply.

Vector Operations

In this problem, you're asked to find the dot product, one of the key operations with vectors.
The dot product of vectors \( \mathbf{v} = \left\langle \frac{3}{2}, -1 \right\rangle \) and \( \mathbf{w} = \langle 4, 0 \rangle \) is a form of vector multiplication resulting in a scalar.
Steps to compute the dot product are straightforward:

• Multiply corresponded components: For the given vectors, you'll multiply \( \frac{3}{2} \) by \( 4 \) and \( -1 \) by \( 0 \).
• Sum the results: Add the two products: \( 6 + 0 \) gives the dot product, which is \( 6 \).

This operation helps find projections and angles between vectors. Its simplicity makes it a powerful tool across various applications, such as determining orthogonal vectors when the dot product is zero.

Precalculus

Precalculus serves as the bridge between basic algebra and calculus, often incorporating the study of vectors and operations such as the dot product.
This subject builds foundational understanding necessary for tackling calculus questions and real-world problems involving multidimensional data.

• Vector Operations: An emphasis is placed on mastering operations like addition, subtraction, and dot product of vectors, as seen in your given problem.
• Spatial Reasoning: It develops your capacity to visualize problem situations in 2D or 3D space.
  Such skills are vital as they lay groundwork for mathematical problem-solving involving motion and optimization.

Grasping the concept of the dot product, as in the provided exercise, prepares you for further mathematical studies in calculus and beyond, where you will delve deeper into the intricacies of vector calculus.