108新課綱｜高二數學｜【延伸】垂直向量的判定及應用

Summary

This script delves into the principles of facial recognition technology, highlighting the role of vectors derived from facial features. It explains how the comparison of vectors, such as the one from the nose to the corner of the mouth, helps in recognizing individuals despite variations due to angles or expressions. The script further explores vector operations, including calculating the angle between vectors using dot product and magnitude, emphasizing the significance of the cosine value in determining the angle. It concludes with the concept of perpendicular vectors, where the dot product equals zero, and applies this principle to prove that the angle in a semicircle is a right angle, offering an engaging mathematical exploration for viewers.

Takeaways

• 📸 Facial recognition technology utilizes vectors derived from key facial points such as the nose, mouth, and eyes.
• 🔍 The process involves comparing vectors like the one from the nose to the mouth, ensuring consistency across different scans.
• 🔄 Variations in scanning angles or facial expressions can introduce slight differences in vectors, but small deviations are acceptable.
• 🧮 The angle between two vectors can be calculated using the dot product formula: (a1*b1 + a2*b2) / (|a| * |b|).
• 💡 If the angle between vectors a and b is 90 degrees, they are perpendicular, meaning their dot product is zero.
• 📏 The dot product's relation to angles helps in determining if vectors are perpendicular or if their angle is acute or obtuse.
• 📐 When two vectors are perpendicular, the dot product is zero; if the angle is acute, the dot product is positive; if obtuse, it's negative.
• 🔢 The concept of vector perpendicularity and dot products can be applied to prove geometric properties, such as angles in circles.
• 🏆 A key geometric proof using vectors shows that an angle inscribed in a semicircle is always a right angle.
• 🧩 This proof involves showing that the dot product of vectors from the circle's center to any point on the semicircle is zero.

Q & A

What is the principle of facial recognition mentioned in the script?

The script mentions that one of the principles of facial recognition involves using vectors derived from facial features such as the center of the nose, corners of the mouth, and corners of the eyes, along with their coordinates.

How does the phone record the initial facial recognition vectors?

The phone records the initial vectors, such as the vector from the center of the nose to the corner of the mouth, during the first setup of facial recognition.

What is the significance of vector 'a' in facial recognition?

Vector 'a' represents the vector from the center of the nose to the corner of the mouth. It is used as a reference to identify the user in subsequent facial recognition scans.

Why might there be a difference between the initial vector 'a' and a subsequent vector 'a''?

Differences can occur due to changes in the angle of the photo or variations in facial expressions, which can cause a slight difference between the initial and subsequent vectors.

What is the acceptable difference in the angle between vectors 'a' and 'a'' for successful recognition?

The acceptable difference is when the angle is less than a certain predefined value, which is considered an acceptable variation for recognition.

How is the angle between two vectors calculated?

The angle can be calculated using the dot product of the vectors and the magnitudes of the vectors, applying the formula for cosine of the angle θ.

What is the dot product of two vectors and how is it related to the angle between them?

The dot product of two vectors is calculated as the product of their magnitudes and the cosine of the angle between them. It is directly related to the angle, as it helps determine the cosine of the angle.

What does it mean for two vectors to be perpendicular?

Two vectors are perpendicular if their angle is 90 degrees. In this case, the dot product of the vectors is zero.

How can you determine if two vectors are perpendicular using their dot product?

If the dot product of two non-zero vectors is zero, it indicates that the vectors are perpendicular to each other.

What is the relationship between the dot product and the angle when the angle is acute?

When the angle θ is acute, the cosine of the angle is positive, and thus the dot product is a positive number.

What is the relationship between the dot product and the angle when the angle is obtuse?

When the angle θ is obtuse, the cosine of the angle is negative, resulting in a negative dot product.

How can the concept of vectors and their dot product be used to prove geometric properties?

The script provides an example of using the dot product to prove that the angle in a semicircle with diameter AB at point P is 90 degrees by showing that the vectors PA and PB are perpendicular.