Problem 5 - Algebra and Trigonometry

Question

Use the given vectors to find \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{v} \cdot \mathbf{v}\). $$ \mathbf{v}=-6 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=-10 \mathbf{i}-8 \mathbf{j} $$

Step-by-Step Solution

Answer

The dot product \(\mathbf{v} \cdot \mathbf{w}\) is 100 and the dot product \(\mathbf{v} \cdot \mathbf{v}\) is 61.

Step 1: Identify the Components of the Vectors

From the given equations, we can identify that the components of vector \(\mathbf{v}\) are -6 and -5, and those of vector \(\mathbf{w}\) are -10 and -8.

Step 2: Calculate \(\mathbf{v} \cdot \mathbf{w}\)

The dot product \(\mathbf{v} \cdot \mathbf{w}\) can be calculated by multiplying corresponding components of \(\mathbf{v}\) and \(\mathbf{w}\) and then adding them. So, \(\mathbf{v} \cdot \mathbf{w}\) = (-6*-10) + (-5*-8) = 60 + 40 = 100.

Step 3: Calculate \(\mathbf{v} \cdot \mathbf{v}\)

The dot product \(\mathbf{v} \cdot \mathbf{v}\) can be calculated similarly by multiplying corresponding components of \(\mathbf{v}\) and \(\mathbf{v}\) and then adding them. So, \(\mathbf{v} \cdot \mathbf{v}\) = (-6*-6) + (-5*-5) = 36 + 25 = 61.

Key Concepts

Vectors in Algebra

In algebra, vectors are fundamental elements that represent both magnitude and direction. They can be thought of as arrows pointing from one location to another, and they are often used in physics, engineering, and computer science to model forces, velocities, and other directional quantities.

Vectors are typically written in a component form, such as \(\mathbf{v} = a \mathbf{i} + b \mathbf{j}\), where \(a\) and \(b\) denote the vector's magnitude along the \(x\)-axis and \(y\)-axis respectively, while \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors along these two axes. In the exercise, the vectors are given in this format, enabling us to easily identify and manipulate their components to perform various algebraic operations such as addition, subtraction, and particularly the dot product.

Calculating Dot Products

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. This operation is key in many areas of mathematics and physics because it intimately relates to the cosine of the angle between two vectors and therefore to various methods of calculating work, projecting vectors, and more.

To calculate the dot product of two vectors, you multiply the corresponding components of each vector and then sum up these products. Mathematically, for vectors \(\mathbf{v}\) and \(\mathbf{w}\) with components \(\mathbf{v} = (v_x, v_y)\) and \(\mathbf{w} = (w_x, w_y)\), the dot product is \(\mathbf{v} \cdot \mathbf{w} = v_x \times w_x + v_y \times w_y\). It’s important to understand that this operation produces a scalar result, hence the term 'scalar product'.

Vector Components

Vector components are an expression of a vector in terms of its effect in multiple directions—typically represented along the axes of whichever coordinate system is being used. For two-dimensional vectors, you will generally see components along the \(x\)-axis (horizontal) and the \(y\)-axis (vertical).

Each component of a vector acts independently and can be dealt with as a separate quantity while performing vector algebra. This makes understanding and visualizing vectors and their operations much easier. When you see a vector written like \(\mathbf{v}=-6 \mathbf{i}-5 \mathbf{j}\), the numbers -6 and -5 are the magnitudes of that vector's components along the \(x\)-axis and \(y\)-axis, respectively, and because they are negative, they indicate directions opposite to the positive orientation of these axes. Recognizing each component is essential in executing vector operations accurately, as vividly demonstrated in the exercise provided.