If `vecrxxvecb=veccxxvecb` and `vecr_|_veca` then `vecr` is equal to

Options

A. `(vecaxx(veccxxvecb))/(veca.vecb)`
B. `(vecbxx(vecaxxvecc))/(veca.vecb)`
C. `(veccxx(vecaxxvecb))/(veca.vecb)`
D. `(veccxx(vecaxxvecb))/(vecb.vecc)`

Explanation

To solve the problem, we need to find the vector \(\vec{r}\) given the conditions:

1. \(\vec{r} \times \vec{b} = \vec{c} \times \vec{b}\)
2. \(\vec{r} \perp \vec{a}\)

Let's break down the solution step by step.

### Step 1: Use the Cross Product Condition

From the first condition, we can rearrange it as follows:

\[ \vec{r} \times \vec{b} - \vec{c} \times \vec{b} = 0 \]

This implies:

\[ \vec{r} \times \vec{b} = \vec{c} \times \vec{b} \]

### Step 2: Apply the Properties of Cross Products

Using the property of cross products, we can factor out \(\vec{b}\):

\[ (\vec{r} - \vec{c}) \times \vec{b} = 0 \]

This means that \(\vec{r} - \vec{c}\) is parallel to \(\vec{b}\). Therefore, we can express \(\vec{r}\) in terms of \(\vec{c}\) and \(\vec{b}\):

\[ \vec{r} = \vec{c} + \lambda \vec{b} \]

where \(\lambda\) is some scalar.

### Step 3: Use the Perpendicular Condition

From the second condition, since \(\vec{r} \perp \vec{a}\), we have:

\[ \vec{r} \cdot \vec{a} = 0 \]

Substituting \(\vec{r}\) from the previous step:

\[ (\vec{c} + \lambda \vec{b}) \cdot \vec{a} = 0 \]

### Step 4: Expand the Dot Product

Expanding the dot product gives us:

\[ \vec{c} \cdot \vec{a} + \lambda (\vec{b} \cdot \vec{a}) = 0 \]

### Step 5: Solve for \(\lambda\)

Rearranging the equation to solve for \(\lambda\):

\[ \lambda = -\frac{\vec{c} \cdot \vec{a}}{\vec{b} \cdot \vec{a}} \]

### Step 6: Substitute \(\lambda\) Back into the Expression for \(\vec{r}\)

Now, substituting \(\lambda\) back into the expression for \(\vec{r}\):

\[ \vec{r} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{\vec{b} \cdot \vec{a}} \vec{b} \]

### Step 7: Rewrite the Expression

This can be rewritten as:

\[ \vec{r} = \vec{c} + \left(-\frac{\vec{c} \cdot \vec{a}}{\vec{b} \cdot \vec{a}}\right) \vec{b} \]

### Final Expression

Thus, we have expressed \(\vec{r}\) in terms of \(\vec{c}\) and \(\vec{b}\):

\[ \vec{r} = \vec{c} + \frac{-\vec{c} \cdot \vec{a}}{\vec{b} \cdot \vec{a}} \vec{b} \]