How to plot the following: Find the angle between the vectors 2i + 3j and i - j.

To find the angle between the vectors \( \mathbf{a} = 2\mathbf{i} + 3\mathbf{j} \) and \( \mathbf{b} = \mathbf{i} - \mathbf{j} \), use the formula:

\[
\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}
\]

1. Calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \):
   \[
   \mathbf{a} \cdot \mathbf{b} = (2)(1) + (3)(-1) = 2 - 3 = -1
   \]

2. Calculate the magnitudes:
   \[
   |\mathbf{a}| = \sqrt{(2^2 + 3^2)} = \sqrt{4 + 9} = \sqrt{13}
   \]
   \[
   |\mathbf{b}| = \sqrt{(1^2 + (-1)^2)} = \sqrt{1 + 1} = \sqrt{2}
   \]

3. Substitute these values into the cosine formula:
   \[
   \cos \theta = \frac{-1}{\sqrt{13} \cdot \sqrt{2}} = \frac{-1}{\sqrt{26}}
   \]

4. Finally, find the angle:
   \[
   \theta = \cos^{-1}\left(\frac{-1}{\sqrt{26}}\right)
   \]

You can plot the vectors in a coordinate system, showing their directions and the angle between them.

Would you like any additional details or help with something else? Here are some related questions you might consider:

1. How do you derive the formula for the angle between two vectors?
2. Can you explain how to find the unit vector of a given vector?
3. What are some applications of vector angles in physics?
4. How do you visualize vectors in a 3D space?
5. What are other methods to find the angle between two vectors? 

**Tip:** Always sketch the vectors to better understand their orientation and the resulting angle.

Discover how to find the angle between the vectors 2i + 3j and i - j using the dot product and magnitudes. This detailed guide covers essential vector concepts and step-by-step calculations, 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