Problem 43 - Precalculus

Question

Find a unit vector in the direction of the given vector. $$\mathbf{v}=\langle\sqrt{2}, 3 \sqrt{2}\rangle$$

Step-by-Step Solution

Answer

The unit vector is \( \left\langle \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right\rangle. \)

Step 1: Understanding the Given Vector

We are presented with a vector \( \mathbf{v} = \langle \sqrt{2}, 3 \sqrt{2} \rangle \). Our goal is to find a unit vector in the same direction as this vector. A unit vector is a vector with a magnitude (or length) of 1.

Step 2: Calculating the Magnitude of the Vector

The magnitude of vector \( \mathbf{v} = \langle \sqrt{2}, 3 \sqrt{2} \rangle \) is calculated using the formula \( ||\mathbf{v}|| = \sqrt{x^2 + y^2} \). Substitute the components into the formula to get:\[ ||\mathbf{v}|| = \sqrt{(\sqrt{2})^2 + (3\sqrt{2})^2} = \sqrt{2 + 18} = \sqrt{20} = 2\sqrt{5}. \]

Step 3: Finding the Unit Vector

Once we have the magnitude of the original vector, \( 2\sqrt{5} \), the unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is obtained by dividing each component of \( \mathbf{v} \) by its magnitude:\[ \mathbf{u} = \left\langle \frac{\sqrt{2}}{2\sqrt{5}}, \frac{3\sqrt{2}}{2\sqrt{5}} \right\rangle = \left\langle \frac{\sqrt{2}}{2\sqrt{5}}, \frac{3\sqrt{2}}{2\sqrt{5}} \right\rangle. \]Simplifying, we get:\[ \mathbf{u} = \left\langle \frac{\sqrt{2}}{2\sqrt{5}}, \frac{3\sqrt{2}}{2\sqrt{5}} \right\rangle. \]

Step 4: Simplifying the Unit Vector's Components

Let's simplify further using the property \( \frac{\sqrt{2}}{2\sqrt{5}} = \frac{\sqrt{2}}{2\cdot\sqrt{5}} = \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{10}}{10} \). Repeat for both components:- The first component simplifies to \( \frac{1}{\sqrt{10}} \) by multiplying the numerator and the denominator by \( \sqrt{5} \) and simplifying.- The second component simplifies to \( \frac{3}{\sqrt{10}} \) similarly.Finally, we get:\[ \mathbf{u} = \left\langle \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right\rangle. \]

Key Concepts

vector magnitude

The magnitude of a vector is akin to measuring its length. For a vector with components, such as \(\mathbf{v} = \langle x, y \rangle\), you can visualize it as an arrow pointing in space.
To discover its length, use the Pythagorean theorem to compute:

• \( || \mathbf{v} || = \sqrt{x^2 + y^2} \)

This formula helps you calculate how far that 'arrow' stretches from its starting to its end point.
For our specific example, the vector \(\mathbf{v} = \langle \sqrt{2}, 3 \sqrt{2} \rangle\), its magnitude becomes:
\[ || \mathbf{v} || = \sqrt{(\sqrt{2})^2 + (3\sqrt{2})^2} = \sqrt{2 + 18} = \sqrt{20} = 2\sqrt{5} \]Understanding this is crucial since it offers a foundation to convert any vector to a unit vector.

vector components

The term vector components refer to the parts that make up a vector. Imagine a vector as a journey: each component points in a particular direction, similar to taking steps east and then north.
For a 2D vector like \(\mathbf{v} = \langle x, y \rangle\):

• \(x\) is how far it extends horizontally,
• \(y\) is its vertical reach.

In our example \(\mathbf{v} = \langle \sqrt{2}, 3 \sqrt{2} \rangle\), \(\sqrt{2}\) is the horizontal component, whereas \(3\sqrt{2}\) is the vertical one.
Understanding vector components enables you to manipulate and work with vectors in various calculations, particularly when you're transforming vectors to solve real-world problems. This understanding is key in finding where vectors 'point.'

unit vector calculation

The goal of unit vector calculation is straightforward: transform any given vector into a vector that maintains the same direction but measures exactly one unit in length.
To achieve this with our vector \(\mathbf{v}\):

• First, compute the vector's magnitude (as explained earlier).
• Next, divide each component of the vector \(\mathbf{v} = \langle x, y \rangle\) by this magnitude.

For \(\mathbf{v} = \langle \sqrt{2}, 3\sqrt{2} \rangle\) with magnitude \(2\sqrt{5}\), the calculation is:
\[ \mathbf{u} = \left\langle \frac{\sqrt{2}}{2\sqrt{5}}, \frac{3\sqrt{2}}{2\sqrt{5}} \right\rangle \]You further simplify:
\[ \mathbf{u} = \left\langle \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right\rangle \]This new vector \(\mathbf{u}\) keeps the direction but has a magnitude of 1; hence, it is a unit vector. By mastering unit vector calculations, you gain the ability to normalize vectors consistently, a skill useful in many fields such as physics and engineering.