`phati+3hatj+4hatk and sqrt(q)i+4hatk` are two vectors, where `p,q ge 0` are two scalars, the length of the vectors are equal then

Options

A. all values of (p,q)
B. only finite number of values of (p,q)
C. infinite number of values (p,q)
D. no value of (p,q)

Answer

To solve the problem, we need to find the values of the scalars \( p \) and \( q \) such that the lengths (magnitudes) of the two given vectors are equal.

### Step 1: Define the vectors
Let vector \( \mathbf{a} = p \hat{i} + 3 \hat{j} + 4 \hat{k} \) and vector \( \mathbf{b} = \sqrt{q} \hat{i} + 4 \hat{k} \).

### Step 2: Calculate the magnitudes of the vectors
The magnitude of vector \( \mathbf{a} \) is given by:
\[ |\mathbf{a}| = \sqrt{p^2 + 3^2 + 4^2} = \sqrt{p^2 + 9 + 16} = \sqrt{p^2 + 25} \]

The magnitude of vector \( \mathbf{b} \) is given by:
\[ |\mathbf{b}| = \sqrt{(\sqrt{q})^2 + 0^2 + 4^2} = \sqrt{q + 16} \]

### Step 3: Set the magnitudes equal
Since the lengths of the vectors are equal, we have:
\[ |\mathbf{a}| = |\mathbf{b}| \]
This leads to the equation:
\[ \sqrt{p^2 + 25} = \sqrt{q + 16} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ p^2 + 25 = q + 16 \]

### Step 5: Rearrange the equation
Rearranging the equation results in:
\[ p^2 - q + 9 = 0 \]
This can be rewritten as:
\[ q = p^2 + 9 \]

### Step 6: Analyze the conditions for \( p \) and \( q \)
Since \( p \) and \( q \) are scalars that must be non-negative (\( p, q \geq 0 \)), we analyze the equation \( q = p^2 + 9 \):
- The minimum value of \( p^2 \) is \( 0 \) (when \( p = 0 \)), which gives \( q = 9 \).
- As \( p \) increases, \( q \) also increases without bound.

### Step 7: Conclusion
Thus, we find that:
- \( q \) must be greater than or equal to \( 9 \) (i.e., \( q \geq 9 \)).
- There are infinite pairs of \( (p, q) \) that satisfy the equation since for every non-negative value of \( p \), there is a corresponding value of \( q \).

### Final Answer
The values of \( p \) and \( q \) can take infinitely many pairs such that \( q = p^2 + 9 \) with the condition \( q \geq 9 \).