A

Step 1: Let's denote the magnitudes of the two vectors as \( A \) and \( B \). When two vectors are at right angles to each other, the resultant magnitude can be calculated using the Pythagorean theorem:
\( R = \sqrt{A^2 + B^2} \).
Step 2: According to the problem, the maximum resultant is 17 and the minimum resultant is 7. This leads to the equations:
\( A + B = 17 \) (maximum)
\( |A - B| = 7 \) (minimum)
Step 3: Now, we can express \( A \) and \( B \) using these equations. From the second equation, we can derive two cases:
Case 1: \( A - B = 7 \)
Case 2: \( B - A = 7 \)
Step 4: Let's solve Case 1:
From this, we have:
\( A = B + 7 \).
Substituting into the first equation:
\( (B + 7) + B = 17 \)
\( 2B + 7 = 17 \)
\( 2B = 10 \)
\( B = 5 \) and \( A = 12 \).
Step 5: In Case 2,
\( B = A + 7 \) gives us:
\( A + (A + 7) = 17 \)
Which leads to no valid solution since the vectors cannot be negative.
Step 6: Thus, we have the magnitudes of the vectors as 12 and 5.
Step 7: Now, when they are at right angles to each other, we find the resultant:
\( R = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \).
Therefore, the correct answer is confirmed to be 13.