Problem 7 - Precalculus with Limits

Question

In Exercises 7-14, find the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u} = \langle 7, 1 \rangle\) \(\mathbf{v} = \langle -3, 2 \rangle\)

Step-by-Step Solution

Answer

The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is -19.

Step 1: Identify the components of given vectors

From the given vectors, we can identify the components for vector \(\mathbf{u}\) as \(7, 1\) and for vector \(\mathbf{v}\) as \(-3, 2\). So \(u_1 = 7, u_2 = 1, v_1 = -3\) and \(v_2 = 2\).

Step 2: Apply the formula for dot product

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is calculated as \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 = (7)(-3) + (1)(2)\).

Step 3: Evaluate the expression

After multiplying and then adding, we get \(\mathbf{u} \cdot \mathbf{v} = -21 + 2 = -19\).

Key Concepts

Vector Operation

When we talk about vector operations, we refer to various calculations that can be performed with vectors. Vectors are not just quantities with magnitude; they also have direction, which makes computations with them fundamentally different than those with scalar quantities, which have only magnitude.

One of the most fundamental vector operations is the dot product, also known as the scalar product because it results in a scalar value. The dot product between two vectors is a measure of their similarity in direction and is calculated by multiplying the corresponding components of each vector and then summing these products. It's given by the formula \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n \) for vectors in n-dimensional space.

A positive dot product indicates that the vectors are pointed in a generally similar direction, while a negative value suggests they are pointed in more opposite directions. If the dot product equals zero, the vectors are orthogonal (perpendicular to each other). This concept is essential in various fields such as physics, where it can, for instance, indicate whether the work done is positive, negative, or zero.

Algebraic Vectors

Algebraic vectors are representations of vectors in a format that makes them easy to work with in algebraic computations. Unlike geometric vectors, which are often visualized by arrows in space, algebraic vectors are expressed in terms of their components which correspond to coordinates in a geometrical space, such as \(\mathbb{R}^2\) (the plane) or \(\mathbb{R}^3\) (three-dimensional space).

The notation for an algebraic vector in two dimensions is \(\mathbf{v} = \langle v_1, v_2 \rangle\), which identifies its position relative to a coordinate system. This form is particularly helpful for computational purposes, allowing for straightforward calculations of operations such as addition, subtraction, scaling (multiplication by a scalar), and the dot product. Algebraic vectors make abstract vector concepts more concrete and applicable to problem-solving in mathematics, engineering, and science.

Vector Components

The components of a vector are the projections of that vector onto the axes of the coordinate system. In a two-dimensional space, a vector \(\mathbf{v}\) has two components, \(v_1\) and \(v_2\), which represent how far the vector extends along the x-axis and y-axis, respectively. These components allow us to analyze vectors algebraically by performing calculations on each part separately.

The significance of vector components cannot be overstated as they are central to many vector operations. When you find the dot product, for example, it's the components of each vector you multiply and add together. Understanding the role of components is crucial for moving between the geometric and algebraic representations of vectors and for grasping the essence of vector manipulations.