Question

If \( AB = 2i + 3j - 6k \), \( BC = 6i - 2j + 3k \) are the vectors along two sides of a triangle ABC, then the perimeter of triangle ABC is:

• A. 21
• B. \( \sqrt{74} + 14 \)
• C. \( \sqrt{74} + 19 \)
• D. \( \sqrt{74} + 3 \)

Hint:

To find the perimeter of a triangle given vectors, calculate the magnitude of each side and sum them.

Answer 1

The Correct Option is B

To find the perimeter of triangle \( ABC \), we first determine the lengths of the sides \( AB \), \( BC \), and \( CA \). The perimeter is the sum of these lengths. Step 1: Find the length of \( AB \) The vector \( AB \) is given by: \[ AB = 2i + 3j - 6k. \] The length of \( AB \) is: \[ |AB| = \sqrt{2^2 + 3^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7. \] Step 2: Find the length of \( BC \) The vector \( BC \) is given by: \[ BC = 6i - 2j + 3k. \] The length of \( BC \) is: \[ |BC| = \sqrt{6^2 + (-2)^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7. \] Step 3: Find the length of \( CA \) The vector \( CA \) is the negative of the sum of \( AB \) and \( BC \): \[ CA = -(AB + BC) = -[(2i + 3j - 6k) + (6i - 2j + 3k)] = -(8i + j - 3k) = -8i - j + 3k. \] The length of \( CA \) is: \[ |CA| = \sqrt{(-8)^2 + (-1)^2 + 3^2} = \sqrt{64 + 1 + 9} = \sqrt{74}. \] Step 4: Compute the perimeter The perimeter of triangle \( ABC \) is the sum of the lengths of its sides: \[ \text{Perimeter} = |AB| + |BC| + |CA| = 7 + 7 + \sqrt{74} = 14 + \sqrt{74}. \] Final Answer: \[ \boxed{\sqrt{74} + 14} \]