Problem 22 - APEX Calculus

Question

In Exercises 22-25, find the unit vector \(\vec{u}\) in the direction of \(\vec{v} .\) \(\vec{v}=\langle 3,7\rangle\)

Step-by-Step Solution

Answer

The unit vector is \( \vec{u} = \left\langle \frac{3}{\sqrt{58}}, \frac{7}{\sqrt{58}} \right\rangle.\)

Step 1: Understanding the Concept of a Unit Vector

A unit vector in the direction of a vector \( \vec{v} \) has the same direction as \( \vec{v} \) but a magnitude of 1. To find this unit vector, we must divide \( \vec{v} \) by its magnitude.

Step 2: Calculate the Magnitude of \( \vec{v} \)

The magnitude of a vector \( \vec{v} = \langle a, b \rangle \) is given by \( ||\vec{v}|| = \sqrt{a^2 + b^2} \). Substituting the given values, we have: \[||\vec{v}|| = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}.\]

Step 3: Find the Unit Vector \( \vec{u} \)

We find the unit vector \( \vec{u} \) by dividing each component of \( \vec{v} \) by its magnitude \( ||\vec{v}|| \). Thus, \[\vec{u} = \left\langle \frac{3}{\sqrt{58}}, \frac{7}{\sqrt{58}} \right\rangle.\] This vector has a magnitude of 1 and is in the same direction as \( \vec{v} \).

Step 4: Rewrite the Unit Vector in Simplified Form (Optional)

If required, rationalize the denominator by multiplying the numerator and the denominator by \( \sqrt{58} \). However, it is typically acceptable to leave the unit vector in its current form for practical calculations.

Key Concepts

Vector Magnitude

The magnitude of a vector gives us an idea of how long the vector is, sort of like measuring the "length" of the vector. This is a crucial step when you need to find a unit vector, as you first need to know the vector's magnitude in order to scale it down to a length of one. For a vector \( \vec{v} = \langle a, b \rangle \), the magnitude is calculated using the Pythagorean theorem: \(|\|\vec{v} || = \sqrt{a^2 + b^2}\).
This formula essentially applies the concept that a vector is like the hypotenuse of a right triangle, with its components \( a \) and \( b \) akin to the triangle's sides. By substituting the given values for our vector, \( \vec{v} = \langle 3, 7 \rangle \), we find its magnitude: \(|\|\vec{v}|| = \sqrt{3^2 + 7^2} = \sqrt{58}\).
Knowing this magnitude helps us easily find the vector's unit form by normalizing it, which is just a fancy term for making its length equal to one while keeping the same direction.

Direction of a Vector

The direction of a vector tells us where the vector is heading in the space it occupies. When talking about direction, we usually refer to the angle compared to a horizontal line, or just which way the vector points. When given a vector, say \( \vec{v} = \langle 3, 7 \rangle \), the direction can be determined by two things:

• The ratio between its components.
• Using trigonometric functions to find the angle with respect to a reference line.

This direction is exactly what we preserve when converting our vector into a unit vector. Despite the change in magnitude, the unit vector still points in the same direction. This is why unit vectors are so useful: they give you a direction indicator with just a length of one unit, making mathematical operations more straightforward.

Vector Components

Vector components are the building blocks of a vector. Imagine a vector as an arrow; its components decide how much it goes to the side and how much it goes up or down. The vector \( \vec{v} = \langle 3, 7 \rangle \) has two components:

• The horizontal component \( 3 \)
• The vertical component \( 7 \)

These components tell you exactly how to get from the start of the vector to its end, navigating horizontally and vertically. When finding a unit vector, these components get resized by dividing each one by the vector's magnitude, keeping everything proportional but ensuring the vector now has a total magnitude of one. For example, with \( \vec{v} = \langle 3, 7 \rangle \) and its magnitude \(|\|\vec{v}|| = \sqrt{58}\), the new scaled components become \( \left\langle \frac{3}{\sqrt{58}}, \frac{7}{\sqrt{58}} \right\rangle\), still depicting the same journey in terms of direction, just with mini steps.