Question

The x, y components of vector \( \vec{P} \) have magnitudes 1 and 3 and the x, y components of resultant of \( \vec{P} \) and \( \vec{Q} \) have magnitudes 5 and 6 respectively. What is the magnitude of \( \vec{Q} \)?

• A. 4
• B. 5
• C. 3
• D. 2

Hint:

To find the magnitude of a vector, use the Pythagorean theorem: \( |\vec{V}| = \sqrt{V_x^2 + V_y^2} \).

Answer 1

The Correct Option is B

Step 1: Use of vector addition.
Let the components of \( \vec{P} \) be \( P_x = 1 \) and \( P_y = 3 \), and the components of \( \vec{Q} \) be \( Q_x \) and \( Q_y \). The components of the resultant vector \( \vec{R} = \vec{P} + \vec{Q} \) are given by: \[ R_x = P_x + Q_x = 5, \quad R_y = P_y + Q_y = 6 \] From these, we can find: \[ Q_x = 5 - 1 = 4, \quad Q_y = 6 - 3 = 3 \]
Step 2: Magnitude of \( \vec{Q} \).
The magnitude of \( \vec{Q} \) is: \[ |\vec{Q}| = \sqrt{Q_x^2 + Q_y^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Step 3: Conclusion.
The magnitude of \( \vec{Q} \) is 5, so the correct answer is (B).