Problem 23 - APEX Calculus

Question

Find the unit vector \(\vec{u}\) in the direction of \(\vec{v} .\) \(\vec{v}=\langle 6,8\rangle\)

Step-by-Step Solution

Answer

The unit vector is \( \langle 0.6, 0.8 \rangle \).

Step 1: Understand the Formula for Unit Vector

A unit vector in the direction of a given vector \( \vec{v} \) is calculated by dividing \( \vec{v} \) by its magnitude. The formula is: \[ \vec{u} = \frac{\vec{v}}{\| \vec{v} \|} \] where \( \| \vec{v} \| \) represents the magnitude of the vector \( \vec{v} \).

Step 2: Calculate the Magnitude of the Vector

First, calculate the magnitude of the vector \( \vec{v} = \langle 6, 8 \rangle \) using the formula: \[ \| \vec{v} \| = \sqrt{x^2 + y^2} \] Substitute \( x = 6 \) and \( y = 8 \): \[ \| \vec{v} \| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

Step 3: Divide the Vector by its Magnitude

Now, use the magnitude to find the unit vector \( \vec{u} \) by dividing each component of \( \vec{v} \) by 10: \[ \vec{u} = \frac{1}{10} \langle 6, 8 \rangle = \langle \frac{6}{10}, \frac{8}{10} \rangle = \langle 0.6, 0.8 \rangle \]

Step 4: Conclusion

The unit vector in the direction of \( \vec{v} = \langle 6, 8 \rangle \) is \( \langle 0.6, 0.8 \rangle \).

Key Concepts

Vector Magnitude

Vector magnitude is a fundamental concept in understanding vectors and their properties. The magnitude of a vector is essentially its length or size. It provides a scalar value, representing the distance a vector covers from its tail to its head. To calculate the magnitude of a vector \(\vec{v} = \langle x, y \rangle\), we use the formula:
\[ \| \vec{v} \| = \sqrt{x^2 + y^2} \]
This formula is derived from the Pythagorean theorem, as each component of the vector forms a right triangle. For example, with a vector \(\vec{v} = \langle 6, 8 \rangle \), we find the magnitude by substituting its components:

• \(x = 6\)
• \(y = 8\)

The calculation becomes:
\[ \| \vec{v} \| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]
This tells us the vector \(\vec{v}\) has a magnitude of 10, indicating its length in a coordinate space. Understanding vector magnitude is crucial for various applications in physics, engineering, and computer science, among others.

Vector Normalization

Vector normalization involves converting a non-unit vector into a unit vector while maintaining its direction. A unit vector is a vector with a magnitude of 1, which makes them ideal for representing direction without involving the vector's magnitude. The process of vector normalization is done by dividing the vector by its magnitude.
For any vector \( \vec{v} = \langle x, y \rangle \), the unit vector \( \vec{u} \) is given by:
\[ \vec{u} = \frac{\vec{v}}{\| \vec{v} \|} \]
This formula calculates a vector with the same direction as \( \vec{v} \), and a magnitude of 1. For \( \vec{v} = \langle 6, 8 \rangle \), we first calculated that \( \| \vec{v} \| = 10 \). Thus, the normalized vector or unit vector \( \vec{u} \) becomes:

• \( \vec{u} = \frac{1}{10} \langle 6, 8 \rangle = \langle 0.6, 0.8 \rangle \)

Normalized vectors are especially useful in programming and physics simulations as they provide consistent reference points without scaling factors. Vector normalization is often used to simplify calculations, particularly those involving directions and orientations.

Direction of a Vector

The direction of a vector refers to the path along which the vector acts. It's a crucial aspect of vectors, as it influences how vectors interact with other vectors, components, and forces. The direction is indicated by the vector’s components and can be understood or visualized using angles or directional ratios.
To find the direction of a vector like \(\vec{v} = \langle 6, 8 \rangle\), consider the angle it makes with the positive x-axis, which can be determined using trigonometry. However, with simple vectors in two dimensions, an intuitive grasp can come from visualizing how the components relate:

• The x-component (6) signifies movement or reach along the x-axis.
• The y-component (8) represents reach along the y-axis.

When the vector is normalized, as in \( \langle 0.6, 0.8 \rangle \), only the direction remains since it has a magnitude of 1. This provides a clear, unscaled idea of the vector's direction. Additionally, calculating angles using functions like \( \text{tan}^{-1}\) for \(y/x\) can provide the direction in degrees or radians, further aiding in visualizing the vector's orientation in space.