`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 relationship between the scalars \( p \) and \( q \) given that the lengths (magnitudes) of the two vectors are equal. The vectors are given as:

1. \( \vec{A} = p \hat{i} + 3 \hat{j} + 4 \hat{k} \)
2. \( \vec{B} = \sqrt{q} \hat{i} + 4 \hat{k} \)

### Step 1: Find the magnitude of vector \( \vec{A} \)

The magnitude of a vector \( \vec{A} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by:

\[ |\vec{A}| = \sqrt{a^2 + b^2 + c^2} \]

For vector \( \vec{A} \):

- \( a = p \)
- \( b = 3 \)
- \( c = 4 \)

Thus, the magnitude of \( \vec{A} \) is:

\[ |\vec{A}| = \sqrt{p^2 + 3^2 + 4^2} = \sqrt{p^2 + 9 + 16} = \sqrt{p^2 + 25} \]

### Step 2: Find the magnitude of vector \( \vec{B} \)

For vector \( \vec{B} \):

- \( a = \sqrt{q} \)
- \( b = 0 \) (since there is no \( \hat{j} \) component)
- \( c = 4 \)

Thus, the magnitude of \( \vec{B} \) is:

\[ |\vec{B}| = \sqrt{(\sqrt{q})^2 + 0^2 + 4^2} = \sqrt{q + 16} \]

### Step 3: Set the magnitudes equal to each other

Since the lengths of the vectors are equal, we set their magnitudes equal:

\[ \sqrt{p^2 + 25} = \sqrt{q + 16} \]

### Step 4: Square both sides to eliminate the square roots

Squaring both sides gives us:

\[ p^2 + 25 = q + 16 \]

### Step 5: Rearrange the equation to express \( q \) in terms of \( p \)

Rearranging the equation, we find:

\[ q = p^2 + 25 - 16 \]
\[ q = p^2 + 9 \]

### Step 6: Analyze the conditions for \( p \) and \( q \)

Given that \( p \geq 0 \) and \( q \geq 0 \):

1. Since \( p \geq 0 \), \( p^2 \geq 0 \) implies \( q \geq 9 \).
2. Therefore, \( q \) can take any value greater than or equal to 9 depending on the value of \( p \).

### Conclusion

The relationship between \( p \) and \( q \) is given by:

\[ q = p^2 + 9 \]

This means that for every non-negative value of \( p \), \( q \) will be at least 9 and can take infinitely many values as \( p \) increases.