In a `Delta ABC, " if " vec(AB) = hati - 7hatj + hatk and vec(BC) = 3 hatj + hatj + 2 hatk, " then " | vec(CA)|=`

Options

A. `sqrt(61)`
B. `sqrt(52)`
C. `sqrt(51)`
D. `sqrt(41)`

Answer

To solve the problem, we need to find the magnitude of the vector \(\vec{CA}\) in triangle \(ABC\) given the vectors \(\vec{AB}\) and \(\vec{BC}\).

### Step-by-step Solution:

1. **Identify the Given Vectors:**
- \(\vec{AB} = \hat{i} - 7\hat{j} + \hat{k}\)
- \(\vec{BC} = \hat{i} + \hat{j} + 2\hat{k}\)

2. **Use the Triangle Law of Vectors:**
According to the triangle law, we have:
\[ \vec{AC} = \vec{AB} + \vec{BC} \]
Therefore, we can express \(\vec{CA}\) as:
\[ \vec{CA} = -\vec{AC} \]

3. **Calculate \(\vec{AC}\):**
Substitute the values of \(\vec{AB}\) and \(\vec{BC}\):
\[ \vec{AC} = (\hat{i} - 7\hat{j} + \hat{k}) + (\hat{i} + \hat{j} + 2\hat{k}) \]

4. **Combine the Vectors:**
Now, combine the components:
- For \(\hat{i}\): \(1 + 1 = 2\)
- For \(\hat{j}\): \(-7 + 1 = -6\)
- For \(\hat{k}\): \(1 + 2 = 3\)

Thus,
\[ \vec{AC} = 2\hat{i} - 6\hat{j} + 3\hat{k} \]

5. **Find \(\vec{CA}\):**
Now, we can find \(\vec{CA}\):
\[ \vec{CA} = -\vec{AC} = -2\hat{i} + 6\hat{j} - 3\hat{k} \]

6. **Calculate the Magnitude of \(\vec{CA}\):**
The magnitude of a vector \(\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}\) is given by:
\[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \]
For \(\vec{CA} = -2\hat{i} + 6\hat{j} - 3\hat{k}\):
\[ |\vec{CA}| = \sqrt{(-2)^2 + 6^2 + (-3)^2} \]

7. **Calculate Each Component's Square:**
- \((-2)^2 = 4\)
- \(6^2 = 36\)
- \((-3)^2 = 9\)

Therefore,
\[ |\vec{CA}| = \sqrt{4 + 36 + 9} = \sqrt{49} = 7 \]

### Final Answer:
\[ |\vec{CA}| = 7 \]