`veca*vecb=abs(vecaxxvecb)` then `abs(veca-vecb)` is

Options

A. `sqrt(a^2+b^2+sqrt2ab)`
B. `sqrt(a^2+b^2-sqrt(2ab))`
C. `sqrt(a^2+b^2+sqrt(2ab))`
D. `sqrt(a^2+b^2-sqrt2ab)`

Answer

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

\[ \vec{a} \cdot \vec{b} = |\vec{a} \times \vec{b}| \]

### Step 1: Use the definitions of dot and cross products
We know that:
- The dot product \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\)
- The magnitude of the cross product \(|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta\)

### Step 2: Set the two expressions equal
From the given equation, we can equate the two expressions:

\[ |\vec{a}| |\vec{b}| \cos \theta = |\vec{a}| |\vec{b}| \sin \theta \]

### Step 3: Cancel out the common terms
Assuming \(|\vec{a}|\) and \(|\vec{b}|\) are not zero, we can divide both sides by \(|\vec{a}| |\vec{b}|\):

\[ \cos \theta = \sin \theta \]

### Step 4: Solve for \(\theta\)
This equation implies:

\[ \tan \theta = 1 \]

The angle \(\theta\) for which \(\tan \theta = 1\) is:

\[ \theta = \frac{\pi}{4} \]

### Step 5: Find the value of \(|\vec{a} - \vec{b}|\)
We use the formula for the magnitude of the difference of two vectors:

\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2 \vec{a} \cdot \vec{b} \]

### Step 6: Substitute \(\vec{a} \cdot \vec{b}\)
We substitute \(\vec{a} \cdot \vec{b}\) using the previously derived expression:

\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2 |\vec{a}| |\vec{b}| \cos \left(\frac{\pi}{4}\right) \]

Since \(\cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\):

\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2 |\vec{a}| |\vec{b}| \cdot \frac{1}{\sqrt{2}} \]

### Step 7: Simplify the expression
This simplifies to:

\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - \sqrt{2} |\vec{a}| |\vec{b}| \]

### Step 8: Take the square root
Taking the square root of both sides gives us:

\[ |\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - \sqrt{2} |\vec{a}| |\vec{b}|} \]

### Final Answer
Thus, the final value of \(|\vec{a} - \vec{b}|\) is:

\[ |\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - \sqrt{2} |\vec{a}| |\vec{b}|} \]