A laser security system in a jewelry shop in Mumbai shoots a beam along the line r = (3i + j − k) + λ(2i − j + 2k). What is the distance of this beam from a diamond placed at (4, 3, 2)?

A laser security system in a jewelry shop in Mumbai shoots a beam along the line r = (3i + j − k) + λ(2i − j + 2k). What is the distance of this beam from a diamond placed at (4, 3, 2)?

• A. 3 units
• B. √(10) units
• C. 2 units
• D. √(14)/3 units

Answer: B) √(10) units

Explanation: Let P = (4, 3, 2). Point on line A = (3, 1, −1). AP = i + 2j + 3k. Direction b = 2i − j + 2k. AP × b = i(4+3) − j(2−6) + k(−1−4) = 7i + 4j − 5k. Magnitude = √(49 + 16 + 25) = √90 = 3√(10). Magnitude of b = √(4+1+4) = 3. Distance = |AP × b| / |b| = 3√(10) / 3 = √(10) units.