If vectors `vecA = hati + 2 hatj + 4 hatk and vecB = 5 hati` represent the two sides of a triangle, then the third side of the triangle can have length equal to

Options

A. 6
B. `sqrt56`
C. both (a) and (b)
D. none of the above

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 inequality theorem. The theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

In this case, we will denote the lengths of the sides corresponding to the vectors \(\vec{A}\) and \(\vec{B}\) as follows:

- Length of side corresponding to \(\vec{A}\): \( |\vec{A}| \)
- Length of side corresponding to \(\vec{B}\): \( |\vec{B}| \)
- Length of the third side: \( |\vec{C}| \)

### Step 1: Calculate the magnitudes of \(\vec{A}\) and \(\vec{B}\)

Given:
\[ \vec{A} = \hat{i} + 2\hat{j} + 4\hat{k} \]
\[ \vec{B} = 5\hat{i} \]

The magnitude of \(\vec{A}\) is calculated as:
\[ |\vec{A}| = \sqrt{(1)^2 + (2)^2 + (4)^2} = \sqrt{1 + 4 + 16} = \sqrt{21} \]

The magnitude of \(\vec{B}\) is calculated as:
\[ |\vec{B}| = \sqrt{(5)^2} = \sqrt{25} = 5 \]

### Step 2: Apply the triangle inequality

According to the triangle inequality, we have:
1. \( |\vec{A}| + |\vec{B}| > |\vec{C}| \)
2. \( |\vec{A}| + |\vec{C}| > |\vec{B}| \)
3. \( |\vec{B}| + |\vec{C}| > |\vec{A}| \)

Substituting the magnitudes we calculated:
1. \( \sqrt{21} + 5 > |\vec{C}| \)
2. \( \sqrt{21} + |\vec{C}| > 5 \)
3. \( 5 + |\vec{C}| > \sqrt{21} \)

### Step 3: Solve the inequalities

1. From \( \sqrt{21} + 5 > |\vec{C}| \):
\[ |\vec{C}| < \sqrt{21} + 5 \]

2. From \( \sqrt{21} + |\vec{C}| > 5 \):
\[ |\vec{C}| > 5 - \sqrt{21} \]

3. From \( 5 + |\vec{C}| > \sqrt{21} \):
\[ |\vec{C}| > \sqrt{21} - 5 \]

### Step 4: Combine the inequalities

Now we need to find the range for \( |\vec{C}| \):
- From \( |\vec{C}| < \sqrt{21} + 5 \)
- From \( |\vec{C}| > 5 - \sqrt{21} \)
- From \( |\vec{C}| > \sqrt{21} - 5 \)

Calculating the numerical values:
- \( \sqrt{21} \approx 4.58 \)
- \( 5 - \sqrt{21} \approx 0.42 \)
- \( \sqrt{21} - 5 \approx -0.42 \) (which is not a valid length)

Thus, we can conclude that:
\[ |\vec{C}| > 0.42 \]
and
\[ |\vec{C}| < 9.58 \quad (\text{since } \sqrt{21} + 5 \approx 9.58) \]

### Final Result

The length of the third side can be in the range:
\[ 0.42 < |\vec{C}| < 9.58 \]