If vector `vec(A)=hat(i)+2hat(j)+4hat(k)` and `vec(B)=5hat(i)` represent the two sides of a triangle, then the third side of the triangle can have length equal to

Options

A. `sqrt(56)`
B. `sqrt(21)`
C. 5
D. 6

Answer

To find the possible lengths of the third side of a triangle formed by the vectors \(\vec{A}\) and \(\vec{B}\), we can use the triangle law of vector addition. The third side can be represented as either the sum or the difference of the two vectors.

### Step-by-step Solution:

1. **Identify the vectors**:
Given:
\[ \vec{A} = \hat{i} + 2\hat{j} + 4\hat{k} \]
\[ \vec{B} = 5\hat{i} \]

2. **Calculate the resultant vector when adding the two vectors**:
The resultant vector \(\vec{R_1}\) when adding \(\vec{A}\) and \(\vec{B}\) is:
\[ \vec{R_1} = \vec{A} + \vec{B} = (\hat{i} + 2\hat{j} + 4\hat{k}) + (5\hat{i}) = (1 + 5)\hat{i} + 2\hat{j} + 4\hat{k} = 6\hat{i} + 2\hat{j} + 4\hat{k} \]

3. **Find the magnitude of \(\vec{R_1}\)**:
The magnitude \(|\vec{R_1}|\) is calculated as follows:
\[ |\vec{R_1}| = \sqrt{(6)^2 + (2)^2 + (4)^2} = \sqrt{36 + 4 + 16} = \sqrt{56} \]

4. **Calculate the resultant vector when subtracting the two vectors**:
The resultant vector \(\vec{R_2}\) when subtracting \(\vec{B}\) from \(\vec{A}\) is:
\[ \vec{R_2} = \vec{A} - \vec{B} = (\hat{i} + 2\hat{j} + 4\hat{k}) - (5\hat{i}) = (1 - 5)\hat{i} + 2\hat{j} + 4\hat{k} = -4\hat{i} + 2\hat{j} + 4\hat{k} \]

5. **Find the magnitude of \(\vec{R_2}\)**:
The magnitude \(|\vec{R_2}|\) is calculated as follows:
\[ |\vec{R_2}| = \sqrt{(-4)^2 + (2)^2 + (4)^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \]

6. **Conclusion**:
The possible lengths of the third side of the triangle can be:
\[ \sqrt{56} \quad \text{or} \quad 6 \]

### Final Answer:
The lengths of the third side of the triangle can be equal to \(\sqrt{56}\) or \(6\).