A solution of the vector equation `vecrxxvecb=vecaxxvecb`, where `veca, vecb` are two given vectors is
where `lamda` is a parameter.

Options

A. `vecr=lamdavecb`
B. `vecr=veca+lamdavecb`
C. `vecr=vecb+lamdaveca`
D. `vecr=lamdaveca`

Explanation

To solve the vector equation \( \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \), where \( \vec{a} \) and \( \vec{b} \) are given vectors, we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \]

### Step 2: Move \( \vec{a} \times \vec{b} \) to the left side
Rearranging the equation gives us:
\[ \vec{r} \times \vec{b} - \vec{a} \times \vec{b} = \vec{0} \]

### Step 3: Factor out \( \vec{b} \)
We can factor out \( \vec{b} \) from the left side:
\[ \vec{r} \times \vec{b} - \vec{a} \times \vec{b} = \vec{0} \implies (\vec{r} - \vec{a}) \times \vec{b} = \vec{0} \]

### Step 4: Analyze the cross product
The equation \( (\vec{r} - \vec{a}) \times \vec{b} = \vec{0} \) implies that the vector \( \vec{r} - \vec{a} \) is parallel to \( \vec{b} \). This is because the cross product of two vectors is zero if and only if the vectors are parallel.

### Step 5: Express \( \vec{r} - \vec{a} \) in terms of \( \vec{b} \)
Since \( \vec{r} - \vec{a} \) is parallel to \( \vec{b} \), we can express it as:
\[ \vec{r} - \vec{a} = \lambda \vec{b} \]
where \( \lambda \) is a scalar parameter.

### Step 6: Solve for \( \vec{r} \)
Now, we can solve for \( \vec{r} \):
\[ \vec{r} = \vec{a} + \lambda \vec{b} \]

### Final Solution
Thus, the solution of the vector equation \( \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \) is:
\[ \vec{r} = \vec{a} + \lambda \vec{b} \]

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