https://r2.erweima.ai/mathbot/image/1725887545432vl53qm90.jpg

It looks like you've uploaded a math problem regarding vectors. The questions focus on vector properties and vector product computations. Let's address each part:

### Problem Statement
Given vectors \(\vec{a}\) and \(\vec{b}\):
- \( \|\vec{a}\| = 1 \)
- \( \|\vec{b}\| = 2 \)
- The angle between \(\vec{a}\) and \(\vec{b}\) is \( \frac{2\pi}{3} \)

You need to calculate:
1. \( \| \vec{a} \times \vec{b} \|\)
2. \( \| (2\vec{a} + \vec{b}) \times (\vec{a} + 2\vec{b}) \|\)

### Solution

#### Part 1: Calculate \( \| \vec{a} \times \vec{b} \|\)
The magnitude of the cross product of two vectors \(\vec{a}\) and \(\vec{b}\) is given by:
\[
\|\vec{a} \times \vec{b}\| = \|\vec{a}\| \|\vec{b}\| \sin(\theta)
\]
Plugging in the given values:
\[
\|\vec{a} \times \vec{b}\| = 1 \times 2 \times \sin\left(\frac{2\pi}{3}\right) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}
\]

#### Part 2: Calculate \( \| (2\vec{a} + \vec{b}) \times (\vec{a} + 2\vec{b}) \|\)
The cross product expression can be expanded using the distributive property:
\[
(2\vec{a} + \vec{b}) \times (\vec{a} + 2\vec{b}) = 2\vec{a} \times \vec{a} + 4\vec{a} \times \vec{b} + \vec{b} \times \vec{a} + 2\vec{b} \times \vec{b}
\]
\[
= 0 + 4\vec{a} \times \vec{b} - \vec{b} \times \vec{a} + 0 = 4\vec{a} \times \vec{b} - \vec{b} \times \vec{a} = 4\vec{a} \times \vec{b} + \vec{a} \times \vec{b}
\]
\[
= 5(\vec{a} \times \vec{b})
\]
The magnitude of this vector is:
\[
\|5(\vec{a} \times \vec{b})\| = 5 \|\vec{a} \times \vec{b}\| = 5 \times \sqrt{3} = 5\sqrt{3}
\]

### Results
1. \( \| \vec{a} \times \vec{b} \| = \sqrt{3} \)
2. \( \| (2\vec{a} + \vec{b}) \times (\vec{a} + 2\vec{b}) \| = 5\sqrt{3} \)

Would you like more details on any part of this solution or have any other questions? Here are some further questions you might consider:

1. How does changing the angle between \(\vec{a}\) and \(\vec{b}\) affect the result of the cross product?
2. What are the geometric interpretations of the cross product?
3. How can this information be applied to problems in physics?
4. Can you provide an example where this kind of calculation is relevant in engineering?
5. How would the solution change if the vectors were not in standard position?

**Tip:** Remember that the cross product of two vectors is a vector perpendicular to both original vectors, and its magnitude is related to the area of the parallelogram spanned by these vectors.

Given vectors \(\vec{a}\) and \(\vec{b}\) with specific properties and magnitudes, calculate the dot product \(\vec{a} \cdot \vec{b}\) and the magnitude of the cross product \(\vec{a} \times \vec{b}\), and the magnitude of a more complex cross product involving linear combinations of these vectors.

Learn how to calculate the dot product and the cross product magnitudes for vectors with given magnitudes and angles. This problem involves advanced vector algebra concepts, including the distributive property of the cross product.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation