Problem 106 - Precalculus with Limits

Question

True or False? In Exercises \(103-106,\) determine whether the statement is true or false. Justify your answer. If \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) is a unit vector, then \(a^{2}+b^{2}=1\)

Step-by-Step Solution

Answer

The statement is true. Since \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) is a unit vector, it follows that the sum of the squares of its components equals to 1, i.e., \(a^{2}+b^{2}=1\).

Step 1: Identify the Unit Vector

A unit vector is defined as a vector of length 1. The vector given, \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\), is a unit vector according to the problem statement.

Step 2: Apply the Definition of Unit Vector

The length (or magnitude) of a vector \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) is given by the formula \(\|\mathbf{u}\| = \sqrt{a^{2}+b^{2}}. Given that \(\mathbf{u}\) is a unit vector, its magnitude is 1. Substituting this into the equation we get \(1 = \sqrt{a^{2}+b^{2}}.

Step 3: Confirm the Statement

To determine whether \(a^{2}+b^{2}=1\), we simply square both sides of the previous equation to get rid of the square root. \(1^{2} = (\sqrt{a^{2}+b^{2}})^{2}\) simplifies to \(1 = a^{2}+b^{2}\).

Key Concepts

Vector Magnitude

When we talk about the magnitude of a vector, we refer to the vector's length or size. Consider a vector as an arrow pointing from one point to another in space. The magnitude of that vector is the distance from the start of the arrow to its point.

For two-dimensional vectors like in our exercise, \( \mathbf{u}=a \mathbf{i}+b \mathbf{j} \), we calculate the magnitude by applying the Pythagorean theorem. This theorem relates the sides of a right-angled triangle to each other. Since the components of the vector \( a \mathbf{i} \) and \( b \mathbf{j} \) can be seen as perpendicular sides of a right triangle, the magnitude is the length of the hypotenuse, given by the formula \[ \|\mathbf{u}\| = \sqrt{a^2 + b^2} \].

The magnitude is always a non-negative number and is instrumental in physics and engineering for describing quantities that have both direction and size, like velocity or force. Understanding the concept of vector magnitude is essential before moving on to more complex operations involving vectors.

Unit Vector Definition

A unit vector is often mentioned in physics and engineering, but what does it actually mean? In simple terms, a unit vector is a vector that has a magnitude of exactly one. It is the standard measure for direction and does not carry any information about magnitude other than its unity.

This property makes unit vectors extremely valuable because they can represent direction without affecting the size of the quantities they describe. Once you have a unit vector, you can scale it up to any size by simply multiplying it by a scalar (a real number), and the direction remains unchanged.

In the problem we're looking at, the statement \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) being a unit vector means that the magnitude of this vector must be one, which leads us to the equation \[ \sqrt{a^2 + b^2} = 1 \], as shown in the step-by-step solution. This property is foundational and allows us to quickly understand and manipulate vector direction in various mathematical and physical contexts.

Vector Components

Vector components are essentially the building blocks of a vector. They describe how much of the vector is moving in each direction. For two-dimensional vectors, such as the one in our exercise, we typically define the components along the horizontal axis (x-axis) and the vertical axis (y-axis). These components are represented by \( a \mathbf{i} \) and \( b \mathbf{j} \) respectively.

Here, \( \mathbf{i} \) and \( \mathbf{j} \) are unit vectors along the x and y axes. The component \( a \) tells us how far along the x-axis the vector moves, while \( b \) gives the distance along the y-axis. To visualize this, you can think of sliding along the x-axis by \( a \) units and then moving parallel to the y-axis by \( b \) units. This movement would end at the head of the vector, thus describing the action of the vector in mathematical terms.

Understanding vector components is crucial when resolving a vector into its constituent parts or when you want to add and subtract vectors. It's the clarity of these components that allows for precision and ease in calculations involving vectors.