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 length of the third side of the triangle formed by the vectors \(\vec{A}\) and \(\vec{B}\), we can follow these steps:

### Step 1: Identify the vectors
The given vectors are:
\[ \vec{A} = \hat{i} + 2\hat{j} + 4\hat{k} \]
\[ \vec{B} = 5\hat{i} \]

### Step 2: Find the resultant vector
The third side of the triangle can be represented by the resultant vector \(\vec{R}\), which is the sum of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{R} = \vec{A} + \vec{B} \]
Substituting the values of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{R} = (\hat{i} + 2\hat{j} + 4\hat{k}) + (5\hat{i}) \]
Combine the like terms:
\[ \vec{R} = (1 + 5)\hat{i} + 2\hat{j} + 4\hat{k} = 6\hat{i} + 2\hat{j} + 4\hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector
The magnitude of the resultant vector \(\vec{R}\) is given by:
\[ |\vec{R}| = \sqrt{(6)^2 + (2)^2 + (4)^2} \]
Calculating each term:
\[ |\vec{R}| = \sqrt{36 + 4 + 16} \]
\[ |\vec{R}| = \sqrt{56} \]

### Step 4: Conclusion
The length of the third side of the triangle is:
\[ \sqrt{56} \]

Thus, the answer is \(\sqrt{56}\).

---