Find vector `vecr` if `vecr.veca=m and vecrxxvecb=vecc,` where `veca.vecb!=0`

Answer

To find the vector \(\vec{r}\) given the equations \(\vec{r} \cdot \vec{a} = m\) and \(\vec{r} \times \vec{b} = \vec{c}\), we can follow these steps:

### Step 1: Start with the given equations
We have two equations:
1. \(\vec{r} \cdot \vec{a} = m\) (Equation 1)
2. \(\vec{r} \times \vec{b} = \vec{c}\) (Equation 2)

### Step 2: Cross Equation 2 with \(\vec{a}\)
We take the cross product of Equation 2 with \(\vec{a}\):
\[ \vec{a} \times (\vec{r} \times \vec{b}) = \vec{a} \times \vec{c} \]

### Step 3: Use the vector triple product identity
Using the vector triple product identity, we can rewrite the left-hand side:
\[ \vec{a} \times (\vec{r} \times \vec{b}) = (\vec{a} \cdot \vec{b}) \vec{r} - (\vec{a} \cdot \vec{r}) \vec{b} \]
Thus, we have:
\[ (\vec{a} \cdot \vec{b}) \vec{r} - (\vec{a} \cdot \vec{r}) \vec{b} = \vec{a} \times \vec{c} \]

### Step 4: Substitute \(\vec{a} \cdot \vec{r}\) with \(m\)
From Equation 1, we know that \(\vec{a} \cdot \vec{r} = m\). Substituting this into our equation gives:
\[ (\vec{a} \cdot \vec{b}) \vec{r} - m \vec{b} = \vec{a} \times \vec{c} \]

### Step 5: Rearranging the equation
Rearranging the equation to solve for \(\vec{r}\):
\[ (\vec{a} \cdot \vec{b}) \vec{r} = m \vec{b} + \vec{a} \times \vec{c} \]

### Step 6: Solve for \(\vec{r}\)
Now, we can isolate \(\vec{r}\):
\[ \vec{r} = \frac{1}{\vec{a} \cdot \vec{b}} (m \vec{b} + \vec{a} \times \vec{c}) \]

### Final Answer
Thus, the vector \(\vec{r}\) is given by:
\[ \vec{r} = \frac{m \vec{b} + \vec{a} \times \vec{c}}{\vec{a} \cdot \vec{b}} \]

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