Given that $\mathbf{u} \cdot \mathbf{v} = 1$, $|\mathbf{u}| = 3$, and $|\mathbf{v}| = 6$, we need to find the value of $2\mathbf{u} \cdot (2\mathbf{u} - 2\mathbf{v})$.

We can approach this step by step:

Step 1: Expand the expression

First, expand the expression:

$2\mathbf{u} \cdot (2\mathbf{u} - 2\mathbf{v}) = 2\mathbf{u} \cdot 2\mathbf{u} - 2\mathbf{u} \cdot 2\mathbf{v}$

This simplifies to:

$4(\mathbf{u} \cdot \mathbf{u}) - 4(\mathbf{u} \cdot \mathbf{v})$

Step 2: Substitute known values

Now, substitute the given values:

• $\mathbf{u} \cdot \mathbf{u} = |\mathbf{u}|^2 = 3^2 = 9$
• $\mathbf{u} \cdot \mathbf{v} = 1$

Substituting these into the expression:

$4(9) - 4(1) = 36 - 4 = 32$

Final Answer

The value of $2\mathbf{u} \cdot (2\mathbf{u} - 2\mathbf{v})$ is $\boxed{32}$.

Do you want details or have any questions?

Here are some related questions:

1. What is the angle between vectors $\mathbf{u}$ and $\mathbf{v}$?
2. How would the result change if $|\mathbf{u}| = 2$ instead?
3. Can you find the value of $\mathbf{u} + \mathbf{v}$ using the given information?
4. What is the geometric interpretation of the dot product in this context?
5. How do you calculate the projection of $\mathbf{u}$ onto $\mathbf{v}$?

Tip: Always check if vectors are perpendicular by verifying if their dot product is zero.