If the projection of `vec(i)+3vec(j)+7vec(k)` on `2vec(i)-3vec(j)+6vec(k)` is ___________.

Answer

To find the projection of the vector \(\vec{a} = \vec{i} + 3\vec{j} + 7\vec{k}\) onto the vector \(\vec{b} = 2\vec{i} - 3\vec{j} + 6\vec{k}\), we can use the formula for the projection of one vector onto another.

### Step-by-step solution:

1. **Identify the vectors:**
\[ \vec{a} = \vec{i} + 3\vec{j} + 7\vec{k} \]
\[ \vec{b} = 2\vec{i} - 3\vec{j} + 6\vec{k} \]

2. **Calculate the dot product \(\vec{a} \cdot \vec{b}\):**
\[ \vec{a} \cdot \vec{b} = (1)(2) + (3)(-3) + (7)(6) \]
\[ = 2 - 9 + 42 = 35 \]

3. **Calculate the magnitude of vector \(\vec{b}\):**
\[ |\vec{b}| = \sqrt{(2)^2 + (-3)^2 + (6)^2} \]
\[ = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

4. **Use the projection formula:**
The projection of \(\vec{a}\) onto \(\vec{b}\) is given by:
\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]

5. **Calculate \(|\vec{b}|^2\):**
\[ |\vec{b}|^2 = 7^2 = 49 \]

6. **Substitute the values into the projection formula:**
\[ \text{proj}_{\vec{b}} \vec{a} = \frac{35}{49} \vec{b} \]
\[ = \frac{5}{7} \vec{b} \]

7. **Substitute \(\vec{b}\) back into the equation:**
\[ = \frac{5}{7} (2\vec{i} - 3\vec{j} + 6\vec{k}) \]
\[ = \frac{10}{7} \vec{i} - \frac{15}{7} \vec{j} + \frac{30}{7} \vec{k} \]

### Final Answer:
The projection of \(\vec{i} + 3\vec{j} + 7\vec{k}\) on \(2\vec{i} - 3\vec{j} + 6\vec{k}\) is:
\[ \frac{10}{7} \vec{i} - \frac{15}{7} \vec{j} + \frac{30}{7} \vec{k} \]