https://r2.erweima.ai/mathbot/image/1725957883357jpbx1yru.jpg

We are given two vectors:

\[
\vec{a} = \hat{i} + 3\hat{j}
\]
\[
\vec{b} = 2\hat{i} - 2\hat{j}
\]

The task is to find the angle between the vectors.

### Formula to find the angle between two vectors:
\[
\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}
\]
Where:
- \(\vec{a} \cdot \vec{b}\) is the dot product of the vectors.
- \( |\vec{a}| \) and \( |\vec{b}| \) are the magnitudes of the vectors.

#### Step 1: Compute the dot product \(\vec{a} \cdot \vec{b}\)
\[
\vec{a} \cdot \vec{b} = (1)(2) + (3)(-2) = 2 - 6 = -4
\]

#### Step 2: Compute the magnitudes \( |\vec{a}| \) and \( |\vec{b}| \)

\[
|\vec{a}| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]
\[
|\vec{b}| = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

#### Step 3: Apply the values in the formula for \( \cos \theta \)
\[
\cos \theta = \frac{-4}{\sqrt{10} \cdot 2\sqrt{2}} = \frac{-4}{2\sqrt{20}} = \frac{-2}{\sqrt{20}}
\]

Simplifying further:
\[
\cos \theta = \frac{-2}{\sqrt{20}} = \frac{-2}{2\sqrt{5}} = \frac{-1}{\sqrt{5}}
\]

Thus, the cosine of the angle is:
\[
\cos \theta = \frac{-1}{\sqrt{5}}
\]

#### Step 4: Find the angle \(\theta\)
To find \(\theta\), take the inverse cosine:
\[
\theta = \cos^{-1}\left( \frac{-1}{\sqrt{5}} \right)
\]

I can compute this numerically if you prefer.

Would you like a numerical answer, or further explanation on any of the steps?

Here are five related questions you might be interested in:
1. How do we compute the dot product of vectors in 3D?
2. What is the geometric interpretation of the dot product?
3. How do we compute the angle between vectors in three dimensions?
4. Can two vectors be orthogonal if their dot product is zero?
5. How do you find the projection of one vector onto another?

**Tip:** For higher-dimensional vectors, the same principles apply for finding angles and dot products; just extend the summation to include all dimensions!

Дано: \(\vec{a} = \hat{i} + 3\hat{j}\), \(\vec{b} = 2\hat{i} - 2\hat{j}\). Найти угол между векторами.

This problem involves finding the angle between the vectors \(\vec{a} = \hat{i} + 3\hat{j}\) and \(\vec{b} = 2\hat{i} - 2\hat{j}\). The solution includes calculating the dot product and vector magnitudes, followed by applying the cosine formula to determine the angle. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation