Question

Let \( P(1,2,3) \) and \( Q(-1,-2,-3) \) be the two points and let \( O \) be the origin. Then \( |\overrightarrow{PQ} + \overrightarrow{OP}| = \)

• A. \( \sqrt{13} \)
• B. \( \sqrt{14} \)
• C. \( \sqrt{24} \)
• D. \( \sqrt{12} \)
• E. \( \sqrt{8} \)

Hint:

Try simplifying vector expressions geometrically — often they reduce to known vectors like \( \vec{OQ} \) or \( \vec{OP} \).

Answer 1

The Correct Option is B

Concept:
• Position vector of a point gives vector from origin.
• Vector between two points: \[ \overrightarrow{PQ} = \vec{Q} - \vec{P} \]
• Vector addition is component-wise.
• Magnitude: \[ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} \]

Step 1: Write position vectors.
\[ \vec{OP} = (1,2,3) \] \[ \vec{OQ} = (-1,-2,-3) \]

Step 2: Find vector \( \overrightarrow{PQ} \).
\[ \overrightarrow{PQ} = \vec{Q} - \vec{P} \] \[ = (-1-1,\,-2-2,\,-3-3) \] \[ = (-2,\,-4,\,-6) \]

Step 3: Interpret \( \overrightarrow{OP} \).
\[ \overrightarrow{OP} = (1,2,3) \] This is already the vector from origin to point \(P\).

Step 4: Add the two vectors.
\[ \overrightarrow{PQ} + \overrightarrow{OP} \] \[ = (-2,-4,-6) + (1,2,3) \] \[ = (-2+1,\,-4+2,\,-6+3) \] \[ = (-1,-2,-3) \]

Step 5: Understand geometrically.
Notice: \[ (-1,-2,-3) = \vec{OQ} \] So effectively: \[ \overrightarrow{PQ} + \overrightarrow{OP} = \overrightarrow{OQ} \] (This is a good conceptual check.)

Step 6: Find magnitude.
\[ |\overrightarrow{OQ}| = \sqrt{(-1)^2 + (-2)^2 + (-3)^2} \] \[ = \sqrt{1 + 4 + 9} \] \[ = \sqrt{14} \]

Step 7: Final Answer.
\[ \boxed{\sqrt{14}} \]