Problem 44 - Contemporary Precalculus

Question

Find a unit vector that has the same direction as \(v\). $$-7 \mathbf{i}+8 \mathbf{j}$$

Step-by-Step Solution

Answer

Question: Find the unit vector that has the same direction as the given vector \(v = -7 \mathbf{i}+8 \mathbf{j}\). Answer: The unit vector with the same direction as \(v\) is \(\mathbf{u} = \frac{-7}{\sqrt{113}}\mathbf{i} + \frac{8}{\sqrt{113}}\mathbf{j}\).

Step 1: Calculate the magnitude of vector \(v\).

To calculate the magnitude of vector \(v\), we use the formula: $$ ||v|| = \sqrt{(-7)^2 + (8)^2} $$ Now, solve for the magnitude: $$ ||v|| = \sqrt{49 + 64} $$ $$ ||v|| = \sqrt{113} $$

Step 2: Find the unit vector.

To find the unit vector in the same direction as \(v\), we just need to divide each component of vector \(v\) by its magnitude. This gives us: $$ \mathbf{u} = \frac{1}{||v||}\mathbf{v} = \frac{1}{\sqrt{113}}(-7\mathbf{i}+8\mathbf{j}) $$ So the unit vector with the same direction as \(v\) is: $$ \mathbf{u} = \frac{-7}{\sqrt{113}}\mathbf{i} + \frac{8}{\sqrt{113}}\mathbf{j} $$

Key Concepts

Vector Magnitude

Vector magnitude is a fundamental concept in mathematics and physics, reflecting the 'size' or 'length' of a vector. In the context of precalculus and vector operations, understanding how to determine vector magnitude is essential. Let's take the vector equation:
\[\[\begin{align*} &\mathbf{v} = -7 \mathbf{i} + 8 \mathbf{j} \end{align*}\]\] The magnitude of this vector, often denoted by double vertical bars as \(||\mathbf{v}||\), is found by taking the square root of the sum of the squares of its components. This can be seen as an application of the Pythagorean theorem in two dimensions, where the components represent the sides of a right-angled triangle and the magnitude is the hypotenuse. Thus, the calculation goes as follows:
\[\[\begin{align*}&\sqrt{(\text{-7})^2 + (\text{8})^2} \end{align*}\]\] Solving this, we arrive at:
\[\[\begin{align*} &\sqrt{49 + 64} \end{align*}\]\] which simplifies to:
\[\[\begin{align*} &\sqrt{113} \end{align*}\]\] This numerical value represents the length of the vector in a two-dimensional space and is a crucial step in finding unit vectors or in performing various vector operations such as vector addition, subtraction, and scalar multiplication.

Vector Operations

Vector operations are operations that can be performed on vectors, including addition, subtraction, scalar multiplication, and finding unit vectors. In our example concerning the unit vector that has the same direction as equation: \[\[\begin{align*} -v = -7\mathbf{i} + 8\mathbf{j} \end{align*}\]\], we focus on scalar division, which is a type of scalar multiplication. Once we have the magnitude \(||\mathbf{v}||\), we can create a unit vector equation: \mathbf{u}\ , which is a vector in the same direction as \mathbf{v}\ but with a magnitude of 1. We achieve this by dividing each component of \mathbf{v}\ by its magnitude, resulting in: equation: \[\[\begin{align*} &\mathbf{u} = \frac{1}{\sqrt{113}}(-7\mathbf{i}+8\mathbf{j}) \end{align*}\]\]. Unit vectors are particularly useful in physics and engineering to indicate direction. Moreover, they're building blocks for more complex vector operations, serving as a basis in vector spaces and facilitating calculations in various applications.

Precalculus

Precalculus serves as the foundation for higher-level mathematics, including calculus. It covers a variety of topics such as functions, algebraic expressions, and, importantly for our discussion, vectors. Grasping precalculus concepts like vector magnitude and operations is paramount for students as they transition to more advanced studies such as calculus and linear algebra. In the study of vectors within the precalculus framework, students learn to visualize and solve problems involving vector addition, scalar multiplication, and finding quantities like magnitudes and unit vectors. The example provided, finding a unit vector from an original vector, is an excellent practical application that bridges the gap between abstract mathematical concepts and real-world problems. Students need to understand the geometric interpretation of vectors and their magnitudes, as well as the arithmetic operations that underpin vector manipulation, to be successful in higher-level mathematics and sciences.