Question

Let \(ABCD\) be a parallelogram. If \( \vec{AB} = \hat{i} + 3\hat{j} + 7\hat{k} \), \( \vec{AD} = 2\hat{i} + 3\hat{j} - 5\hat{k} \), and \( \vec{p} \) is a unit vector parallel to \( \vec{AC} \), then \( \vec{p} \) is equal to

• A. \( \frac{1}{3}(2\hat{i}+\hat{j}+2\hat{k}) \)
• B. \( \frac{1}{3}(2\hat{i}-2\hat{j}+\hat{k}) \)
• C. \( \frac{1}{7}(3\hat{i}+6\hat{j}+2\hat{k}) \)
• D. \( \frac{1}{7}(6\hat{i}+2\hat{j}+3\hat{k}) \)
• E. \( \frac{1}{7}(6\hat{i}+2\hat{j}-3\hat{k}) \)

Hint:

Always remember: diagonal of parallelogram = sum of adjacent sides.

Answer 1

The Correct Option is C

Concept:
• In a parallelogram: \[ \vec{AC} = \vec{AB} + \vec{AD} \]
• A unit vector in direction of vector \( \vec{v} \) is: \[ \frac{\vec{v}}{|\vec{v}|} \]

Step 1: Understand geometry of parallelogram.
In parallelogram \(ABCD\):
• \( \vec{AB} \) and \( \vec{AD} \) are adjacent sides
• Diagonal \( \vec{AC} = \vec{AB} + \vec{AD} \)

Step 2: Write vectors in component form.
\[ \vec{AB} = (1,3,7) \] \[ \vec{AD} = (2,3,-5) \]

Step 3: Find diagonal \( \vec{AC} \).
\[ \vec{AC} = \vec{AB} + \vec{AD} \] \[ = (1+2,\;3+3,\;7-5) \] \[ = (3,6,2) \]

Step 4: Understand direction requirement.
We need a unit vector parallel to \( \vec{AC} \), meaning: \[ \vec{p} = \frac{\vec{AC}}{|\vec{AC}|} \]

Step 5: Find magnitude of \( \vec{AC} \).
\[ |\vec{AC}| = \sqrt{3^2 + 6^2 + 2^2} \] \[ = \sqrt{9 + 36 + 4} \] \[ = \sqrt{49} = 7 \]

Step 6: Find unit vector.
\[ \vec{p} = \frac{1}{7}(3,6,2) \] \[ = \frac{1}{7}(3\hat{i} + 6\hat{j} + 2\hat{k}) \]

Step 7: Match with given options.
This matches: \[ \boxed{\frac{1}{7}(3\hat{i} + 6\hat{j} + 2\hat{k})} \]

Step 8: Final Answer.
\[ \boxed{(C)} \]