Problem 51 - Algebra and Trigonometry

Question

Find the unit vector in the same direction as \(\mathbf{V}\). \(\mathbf{v}=3 \mathbf{i}-4 \mathbf{j}\)

Step-by-Step Solution

Answer

\textbf{u} = \frac{3}{5} \textbf{i} - \frac{4}{5} \textbf{j}.

Step 1 - Find the magnitude of the vector

The magnitude of a vector \(\textbf{v} = 3 \textbf{i} - 4 \textbf{j}\) is found using the formula \[| \textbf{v} | = \sqrt{(3)^2 + (-4)^2}. \] Calculate the magnitude: \[| \textbf{v} | = \sqrt{9 + 16} = \sqrt{25} = 5. \]

Step 2 - Divide the vector by its magnitude

To find the unit vector in the direction of \(\textbf{v}\), divide each component of \(\textbf{v}\) by the magnitude calculated in Step 1. \[ \textbf{u} = \frac{\textbf{v}}{| \textbf{v} |} = \frac{3 \textbf{i} - 4 \textbf{j}}{5}. \]

Step 3 - Simplify each component

Simplify the components of \(\textbf{u}\): \[ \textbf{u} = \left( \frac{3}{5} \textbf{i} - \frac{4}{5} \textbf{j} \right). \] So the unit vector is \(\textbf{u} = \frac{3}{5} \textbf{i} - \frac{4}{5} \textbf{j}\).

Key Concepts

Vector Magnitude

To find a unit vector, the first thing we need to do is determine the magnitude of the original vector. The vector magnitude measures how long the vector is, and it's calculated using the Pythagorean theorem for vectors. Imagine our vector \(\textbf{v} = 3 \textbf{i} - 4 \textbf{j}\). The formula for the magnitude \( |\textbf{v}| = \sqrt{a^2 + b^2}\) is just like finding the hypotenuse of a right triangle: \[ |\textbf{v}| = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] Here, \(a = 3\) and \(b = -4\), representing the components of the vector along the \(i\) and \(j\) directions. Once calculated, the magnitude of our vector is 5. Knowing this gives us a crucial step towards finding the unit vector. Remember, the magnitude is always positive.

Vector Division

Now that we have the magnitude, we need to divide the original vector by this magnitude to get a unit vector. A unit vector has a magnitude of 1 and points in the same direction as the original vector. Let's divide each component of the original vector \(\textbf{v} = 3 \textbf{i} - 4 \textbf{j}\) by its magnitude 5: \[ \textbf{u} = \frac{\textbf{v}}{| \textbf{v} |} = \frac{3 \textbf{i} - 4 \textbf{j}}{5}. \] This division means every component is scaled down proportionally so the resulting vector maintains the direction but with a length of 1. Dividing vectors by their magnitude ensures it shrinks or stretches correctly.

Simplifying Vector Components

Finally, we simplify the equation to make it cleaner and more interpretable. Divide each scalar component of the vector \(3 \textbf{i} - 4 \textbf{j}\) by 5: \[ \textbf{u} = \left( \frac{3}{5} \textbf{i} - \frac{4}{5} \textbf{j} \right). \] This breaks down as: \(\textbf{i}\) component: \(\frac{3}{5}\) and \(\textbf{j}\) component: \(\frac{4}{5}\). Thus, the simplified unit vector is \( \textbf{u} = \frac{3}{5} \textbf{i} - \frac{4}{5} \textbf{j}\). It keeps the directional properties of \(\textbf{v}\) intact while ensuring the length is standardized to 1. Now you have a clean, easy-to-use form of the unit vector.