Let `|A|=1, |vecb|=4` and `veca xxvecr +vecb=vecr`. If the projection of `vecr` along `veca` is 2, then the projection of `vecr` along `vecb` is

Answer

To solve the problem step by step, we will use the given information and properties of vectors.

### Step 1: Understand the Given Information
We have:
- \(|\vec{a}| = 1\)
- \(|\vec{b}| = 4\)
- \(\vec{a} \times \vec{r} + \vec{b} = \vec{r}\)
- The projection of \(\vec{r}\) along \(\vec{a}\) is 2.

### Step 2: Rearranging the Vector Equation
From the equation \(\vec{a} \times \vec{r} + \vec{b} = \vec{r}\), we can rearrange it to find \(\vec{r}\):
\[ \vec{r} = \vec{a} \times \vec{r} + \vec{b} \]

### Step 3: Taking the Dot Product with \(\vec{a}\)
Taking the dot product of both sides with \(\vec{a}\):
\[ \vec{a} \cdot \vec{r} = \vec{a} \cdot (\vec{a} \times \vec{r}) + \vec{a} \cdot \vec{b} \]
Since \(\vec{a} \cdot (\vec{a} \times \vec{r}) = 0\) (the dot product of a vector with a perpendicular vector is zero), we have:
\[ \vec{a} \cdot \vec{r} = \vec{a} \cdot \vec{b} \]

### Step 4: Using the Projection Information
We know that the projection of \(\vec{r}\) along \(\vec{a}\) is 2, which means:
\[ \vec{a} \cdot \vec{r} = 2 \]
Thus, we can conclude:
\[ \vec{a} \cdot \vec{b} = 2 \]

### Step 5: Finding the Projection of \(\vec{r}\) Along \(\vec{b}\)
The projection of \(\vec{r}\) along \(\vec{b}\) is given by the formula:
\[ \text{Projection of } \vec{r} \text{ along } \vec{b} = \frac{\vec{r} \cdot \vec{b}}{|\vec{b}|} \]
We need to find \(\vec{r} \cdot \vec{b}\).

### Step 6: Expressing \(\vec{r}\)
From our rearranged equation, we can express \(\vec{r}\) as:
\[ \vec{r} = \vec{a} \times \vec{r} + \vec{b} \]
Substituting \(\vec{b}\) into the projection formula:
\[ \vec{r} \cdot \vec{b} = (\vec{a} \times \vec{r} + \vec{b}) \cdot \vec{b} \]
This simplifies to:
\[ \vec{r} \cdot \vec{b} = \vec{a} \times \vec{r} \cdot \vec{b} + |\vec{b}|^2 \]
Since \(\vec{a} \times \vec{r} \cdot \vec{b} = 0\) (as \(\vec{a} \times \vec{r}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\)), we have:
\[ \vec{r} \cdot \vec{b} = |\vec{b}|^2 = 4^2 = 16 \]

### Step 7: Calculating the Projection
Now we can calculate the projection:
\[ \text{Projection of } \vec{r} \text{ along } \vec{b} = \frac{16}{|\vec{b}|} = \frac{16}{4} = 4 \]

### Final Answer
The projection of \(\vec{r}\) along \(\vec{b}\) is **4**.

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