Assume that u⋅v=1
, |u|=3
, and |v|=6
.

What is the value of 2u⋅(2u−2v)
?



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.

Assume that u⋅v=1, |u|=3, and |v|=6. What is the value of 2u⋅(2u−2v)?

Solve the vector algebra problem involving dot products and vector magnitudes. Find the value of 2u⋅(2u−2v) using given values u⋅v=1, |u|=3, and |v|=6. This detailed step-by-step solution covers key concepts such as vector algebra, dot products, and the distributive property, suitable for Grades 10-12.

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