If `veca * vecb = |veca xx vecb|`, then this angle between `veca and vecb` is,

Options

A. Zero
B. `pi//2 `
C. `pi//4`
D. `pi`

Answer

To solve the problem, we need to find the angle between two vectors \( \vec{a} \) and \( \vec{b} \) given that:

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

### Step 1: Understand the dot and cross product formulas

The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:

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

where \( \theta \) is the angle between the two vectors.

The magnitude of the cross product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:

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

### Step 2: Set up the equation

According to the problem statement, we have:

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

### Step 3: Simplify the equation

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 \( \theta \)

From the equation \( \cos \theta = \sin \theta \), we can divide both sides by \( \cos \theta \) (as long as \( \cos \theta \neq 0 \)):

\[ 1 = \tan \theta \]

This implies:

\[ \tan \theta = 1 \]

### Step 5: Find the angle \( \theta \)

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

\[ \theta = \tan^{-1}(1) = \frac{\pi}{4} \]

### Conclusion

Thus, the angle between \( \vec{a} \) and \( \vec{b} \) is:

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

### Final Answer

The angle between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{4} \).

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