Problem 35 - Precalculus : Building Concepts and Connections

Question

Find a unit vector in the same direction as the given vector. $$\mathbf{u}=\langle 3,4\rangle$$

Step-by-Step Solution

Answer

The unit vector in the same direction as \(\mathbf{u}\) is \(\langle 0.6, 0.8 \rangle\).

Step 1: Locate the Given Vector

The given vector \(\mathbf{u}\) = \(\langle 3,4\rangle\).

Step 2: Calculate the Magnitude of the Vector

The magnitude of \(\mathbf{u}\) is calculated using the formula \(|\mathbf{u}|\) = \(\sqrt{3^2 + 4^2}\) = \(\sqrt{9 + 16}\) = \(\sqrt{25}\) = 5.

Step 3: Find the Unit Vector

The unit vector \(\mathbf{\hat{u}}\) in the direction of \(\mathbf{u}\) is found by dividing \(\mathbf{u}\) by its magnitude: \(\mathbf{\hat{u}}\) = \(\frac{\mathbf{u}}{|\mathbf{u}|}\) = \(\frac{\langle 3,4 \rangle}{5}\) = \(\langle 0.6, 0.8 \rangle\).

Key Concepts

Vector Magnitude

When we talk about the magnitude of a vector, we are referring to the vector's length or size. It's essentially the distance from the vector's starting point to its endpoint, measured in a straight line. This concept is crucial when working with vectors since it allows us to quantify and compare them. Think of it like measuring how far you'd travel in a straight line from one city to another.

To calculate the magnitude of a vector, we apply the Pythagorean theorem. For a vector \(\mathbf{u}\) with components \(\mathbf{u} = \langle x, y \rangle\), the magnitude is computed using \(\sqrt{x^2 + y^2}\). If we consider the given vector \(\mathbf{u}=\langle 3,4\rangle\), we calculate its magnitude by \(\sqrt{3^2 + 4^2}\) which simplifies to \(\sqrt{9 + 16} = \sqrt{25} = 5\). Knowing how to find a vector's magnitude is foundational, as it's used in various vector operations and is the first step in creating a unit vector.

Vector Operations

Vectors can be manipulated through various operations such as addition, subtraction, scalar multiplication, and dot product. Each operation follows specific rules and has implications for the vectors involved.

For example, adding two vectors \(\mathbf{a}\) and \(\mathbf{b}\) results in a new vector that represents the total displacement if you were to follow \(\mathbf{a}\) then \(\mathbf{b}\). Scalar multiplication involves scaling the vector by a constant, affecting its magnitude but not its direction unless the scalar is negative, which reverses the direction. The dot product, on the other hand, is an operation that combines two vectors to produce a scalar result, often used to determine the angle between the two vectors or to test for orthogonality.

Understanding these basics of vector operations is crucial because it leads to more complex concepts like cross product, vector projection, and, important for our current context, normalizing vectors to find unit vectors.

Normalizing Vectors

Normalizing a vector, also known as finding the unit vector, is the process of scaling a vector so that it retains its direction but its magnitude becomes 1. A unit vector is useful as it often represents a direction and can be used to scale other vectors to that direction.

To normalize a vector \(\mathbf{v}\), you divide each component of \(\mathbf{v}\) by its magnitude \(\|\mathbf{v}\|\). Given our example with \(\mathbf{u}=\langle 3,4\rangle\), after determining the magnitude is 5, we find the unit vector by calculating \(\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|} = \frac{\langle 3,4 \rangle}{5} = \langle 0.6, 0.8 \rangle\).

This unit vector represents the same direction as the original vector but is scaled down so that you can easily use it to determine that direction without worrying about the length of the vector. In physics, for example, this is critical for indicating directions of forces, velocities, or other vector quantities.