If `vec(p)` is a unit vector and `(vec(x)-vec(p)).(vec(x)+vec(p))=80`, then `|vec(x)|=` __________.

Answer

To solve the problem, we start with the given equation:

\[ (\vec{x} - \vec{p}) \cdot (\vec{x} + \vec{p}) = 80 \]

### Step 1: Expand the dot product
Using the distributive property of the dot product, we can expand the left-hand side:

\[ \vec{x} \cdot \vec{x} + \vec{x} \cdot \vec{p} - \vec{p} \cdot \vec{x} - \vec{p} \cdot \vec{p} \]

### Step 2: Simplify the expression
Since the dot product is commutative (\(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\)), we can simplify the expression:

\[ \vec{x} \cdot \vec{x} - \vec{p} \cdot \vec{p} = 80 \]

### Step 3: Substitute the known values
We know that \(\vec{p}\) is a unit vector, which means:

\[ \vec{p} \cdot \vec{p} = 1 \]

Substituting this into the equation gives:

\[ \vec{x} \cdot \vec{x} - 1 = 80 \]

### Step 4: Rearranging the equation
Rearranging the equation to isolate \(\vec{x} \cdot \vec{x}\):

\[ \vec{x} \cdot \vec{x} = 80 + 1 = 81 \]

### Step 5: Find the magnitude of \(\vec{x}\)
The dot product \(\vec{x} \cdot \vec{x}\) is equal to the square of the magnitude of \(\vec{x}\):

\[ |\vec{x}|^2 = 81 \]

Taking the square root of both sides gives:

\[ |\vec{x}| = \sqrt{81} = 9 \]

Thus, the final answer is:

\[ |\vec{x}| = 9 \]

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