To find the angle between the vectors $\mathbf{u} = (6, 5)$ and $\mathbf{v} = (4, 3)$, we can follow the same process as before.

Step 1: Compute the dot product $\mathbf{u} \cdot \mathbf{v}$

$\mathbf{u} \cdot \mathbf{v} = (6)(4) + (5)(3) = 24 + 15 = 39$

Step 2: Compute the magnitudes of $\mathbf{u}$ and $\mathbf{v}$

For $\mathbf{u}$:

$\|\mathbf{u}\| = \sqrt{6^2 + 5^2} = \sqrt{36 + 25} = \sqrt(61)$

For $\mathbf{v}$:

$\|\mathbf{v}\| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Step 3: Compute $\cos(\theta)$

$\cos(\theta) = \frac{39}{\sqrt{61} \times 5}$

Step 4: Compute $\theta$

$\theta = \cos^{-1}\left(\frac{39}{5\sqrt{61}}\right)$

Let's calculate this value to find the angle in degrees.

The angle $\theta$ between the vectors $\mathbf{u} = (6, 5)$ and $\mathbf{v} = (4, 3)$ is approximately $2.9^\circ$.

Would you like further details or explanations about this calculation, or do you have any other questions?

Related Questions:

1. What are the properties of the dot product?
2. How does the angle between vectors affect their dot product?
3. How do you find the angle between vectors in 3D space?
4. What happens to the angle if the vectors are multiplied by a scalar?
5. How do you interpret an angle close to 0° between two vectors?

Tip:

When calculating angles between vectors, a small angle typically indicates that the vectors are nearly aligned.