Problem 45 - Contemporary Precalculus

Question

Find a unit vector that has the same direction as \(v\). $$5 \mathbf{i}+10 \mathbf{j}$$

Step-by-Step Solution

Answer

Question: Find the unit vector in the same direction as the given vector \(5\mathbf{i} + 10\mathbf{j}\). Answer: The unit vector in the same direction as the given vector is \(\hat{v} = \frac{1}{\sqrt{5}}\mathbf{i} + \frac{2}{\sqrt{5}}\mathbf{j}\).

Step 1: Write down the given vector

The given vector is: $$v = 5\mathbf{i} + 10\mathbf{j}$$

Step 2: Calculate the magnitude of the given vector

To calculate the magnitude of the vector \(v\), we can use the formula: $$||v|| = \sqrt{(5^2) + (10^2)} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}$$

Step 3: Find the unit vector

Now that we have the magnitude of the vector \(v\), we can find the unit vector in the same direction by dividing the given vector by its magnitude: $$\hat{v} = \frac{1}{||v||} \cdot v = \frac{1}{5\sqrt{5}} \cdot (5\mathbf{i} + 10\mathbf{j})$$

Step 4: Calculate the unit vector

Now, multiply each component of the vector \(v\) by the scalar \(\frac{1}{5\sqrt{5}}\): $$\hat{v} = \frac{1}{5\sqrt{5}} \cdot 5\mathbf{i} + \frac{1}{5\sqrt{5}} \cdot 10\mathbf{j} = \frac{1}{\sqrt{5}}\mathbf{i} + \frac{2}{\sqrt{5}}\mathbf{j}$$ So, the unit vector that has the same direction as \(v\) is: $$\hat{v} = \frac{1}{\sqrt{5}}\mathbf{i} + \frac{2}{\sqrt{5}}\mathbf{j}$$

Key Concepts

Magnitude of a Vector

Understanding the magnitude of a vector is essential, as it tells us how "long" or "large" the vector is. Think of magnitude as the distance a vector covers in space. For any vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \), the magnitude \( ||\mathbf{v}|| \) is calculated using the Pythagorean theorem. To find this, use the formula:

• \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \)

In the exercise, the vector \( \mathbf{v} \) is \( 5\mathbf{i} + 10\mathbf{j} \). We substitute these values into our formula:

• \( ||\mathbf{v}|| = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \)

Thus, the magnitude of the vector indicates that it stretches across a distance equivalent to \( 5\sqrt{5} \) units in the vectored space. This value is crucial when we later determine the unit vector, ensuring it maintains the same direction but with a standardized magnitude of 1.

Vector Operations

Vector operations include a variety of tasks such as addition, subtraction, and scalar multiplication. In this specific exercise, our focus is scalar multiplication when finding a unit vector. A unit vector has a magnitude of 1 and retains the vector's original direction. To convert a given vector into a unit vector, divide each of its components by the vector's magnitude.For the vector \( \mathbf{v} = 5\mathbf{i} + 10\mathbf{j} \), and its magnitude \( 5\sqrt{5} \), the unit vector \( \hat{v} \) is computed as follows:

• \( \hat{v} = \frac{1}{||\mathbf{v}||} \cdot \mathbf{v} \)
• \( \hat{v} = \frac{1}{5\sqrt{5}} \cdot (5\mathbf{i} + 10\mathbf{j}) \)
• \( \hat{v} = \left( \frac{5}{5\sqrt{5}} \right)\mathbf{i} + \left( \frac{10}{5\sqrt{5}} \right)\mathbf{j} \)
• This simplifies to \( \hat{v} = \frac{1}{\sqrt{5}}\mathbf{i} + \frac{2}{\sqrt{5}}\mathbf{j} \)

This operation ensures the resulting vector has a total magnitude of 1 while maintaining the original vector's trajectory.

Direction of a Vector

The direction of a vector is vital because it tells us the "way" or orientation the vector points. Unlike magnitude, which is about size, direction specifies where the vector is heading in a coordinate system.For example, a vector \( 5\mathbf{i} + 10\mathbf{j} \) has a certain direction regardless of its magnitude. If you think of a vector like an arrow, then the direction is the angle or orientation in which the arrow points.When we calculate a unit vector, although the magnitude changes to 1, the direction remains unchanged. In the exercise, the unit vector \( \hat{v} = \frac{1}{\sqrt{5}}\mathbf{i} + \frac{2}{\sqrt{5}}\mathbf{j} \), ensures that we preserve the original vector's direction. This is because each component of \( \mathbf{i} \) and \( \mathbf{j} \) was individually adjusted by the same scalar, maintaining their ratios.Hence, direction serves as the backbone of vector characteristics, distinguishing how vectors interact spatially within multidimensional spaces and providing clarity on their path and orientation.