https://file.aiquickdraw.com/mathbot/file/1723027269426rbpp8xer.jpg

Dont invent your own vertices, use the position vectors given in the question

Let's solve the given problems using the specified position vectors.

### Given:
The position vectors are:
- \( \vec{a} = 3\hat{i} + 5\hat{j} \)
- \( \vec{b} = \hat{i} - 5\hat{j} \)
- \( \vec{c} = 4\hat{i} + 3\hat{j} \)

### Problem 1: Find \(\angle ABC\)
To find the angle at vertex \(B\), we use the dot product formula. Let's calculate the vectors \(\vec{BA}\) and \(\vec{BC}\):
- \( \vec{BA} = \vec{a} - \vec{b} = (3\hat{i} + 5\hat{j}) - (\hat{i} - 5\hat{j}) = (3 - 1)\hat{i} + (5 - (-5))\hat{j} = 2\hat{i} + 10\hat{j} \)
- \( \vec{BC} = \vec{c} - \vec{b} = (4\hat{i} + 3\hat{j}) - (\hat{i} - 5\hat{j}) = (4 - 1)\hat{i} + (3 - (-5))\hat{j} = 3\hat{i} + 8\hat{j} \)

The dot product \( \vec{BA} \cdot \vec{BC} \) is:
\[ \vec{BA} \cdot \vec{BC} = (2\hat{i} + 10\hat{j}) \cdot (3\hat{i} + 8\hat{j}) = (2 \cdot 3) + (10 \cdot 8) = 6 + 80 = 86 \]

The magnitudes of \(\vec{BA}\) and \(\vec{BC}\) are:
\[ |\vec{BA}| = \sqrt{2^2 + 10^2} = \sqrt{4 + 100} = \sqrt{104} = 2\sqrt{26} \]
\[ |\vec{BC}| = \sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} \]

Using the dot product formula:
\[ \cos \theta = \frac{\vec{BA} \cdot \vec{BC}}{|\vec{BA}| |\vec{BC}|} = \frac{86}{2\sqrt{26} \cdot \sqrt{73}} = \frac{86}{2\sqrt{1898}} = \frac{43}{\sqrt{1898}} \]

Thus, the angle \(\theta\) is:
\[ \theta = \cos^{-1} \left( \frac{43}{\sqrt{1898}} \right) \]

### Problem 2: Verify Pythagorean Theorem
To verify that the triangle is a right triangle, we calculate the squares of the lengths of all sides:
- \( |\vec{AB}| = |\vec{BA}| = 2\sqrt{26} \)
- \( |\vec{BC}| = \sqrt{

Learn how to solve vector geometry problems using position vectors \( \vec{a}, \vec{b}, \vec{c} \) with detailed step-by-step solutions. Explore angles between vectors, verify the Pythagorean theorem, and apply trigonometric formulas. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation