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:
Show Hint
- 21
- \( \sqrt{74} + 14 \)
- \( \sqrt{74} + 19 \)
- \( \sqrt{74} + 3 \)
The Correct Option is B
Solution and Explanation
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: \[ \boxed{\sqrt{74} + 14} \]
Top Questions on Vectors
The respective values of \( |\vec{a}| \) and} \( |\vec{b}| \), if given \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 \quad \text{and} \quad |\vec{a}| = 3 |\vec{b}|, \] are:
- Let \( \vec{a} \) be a position vector whose tip is the point (2, -3). If \( \overrightarrow{AB} = \vec{a} \), where coordinates of A are (–4, 5), then the coordinates of B are:
- Number of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \), that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
- If the sum of two vectors is a unit vector, then the magnitude of their difference is:
- Let A be the point of intersection of the lines $$ L_1 : \frac{x - 7}{1} = \frac{y - 5}{0} = \frac{z - 3}{-1} \quad \text{and} \quad L_2 : \frac{x - 1}{3} = \frac{y + 3}{4} = \frac{z + 7}{5}$$ Let B and C be the points on the lines $ L_1 $ and $ L_2 $, respectively, such that $ AB - AC = \sqrt{15} $. Then the square of the area of the triangle ABC is:
Questions Asked in AP EAMCET exam
- In a meter bridge experiment, a resistance of 9 Ω is connected in the left gap and an unknown resistance greater than 9 Ω is connected in the right gap. If the resistance in the gaps are interchanged, the balancing point shifts by 10 cm. The unknown resistance is:
- AP EAMCET - 2024
- Wheatstone Bridge
- If the longest wavelength of the spectral line of the Paschen series of \( \text{Li}^{2+} \) ion spectrum is \( x \) Å, then the longest wavelength (in Å) of the Lyman series of the hydrogen spectrum is:
- AP EAMCET - 2024
- Atomic Spectra
- Let \( A(2, 3), B(1, -1) \) be two points. If \( P \) is a variable point such that \( \angle APB = 90^\circ \), then the locus of \( P \) is
- AP EAMCET - 2024
- solution of system of linear inequalities in two variables
- The charge \( q \) (in coulombs) passing through a \( 10 \Omega \) resistor as a function of time \( t \) (in seconds) is given by: \[ q = 3t^2 - 2t + 6 \] The potential difference across the ends of the resistor at time \( t = 5 \) s is:
- AP EAMCET - 2024
- coulombs law
The percentage error in the measurement of mass and velocity are 3% and 4% respectively. The percentage error in the measurement of kinetic energy is:
- AP EAMCET - 2024
- Projectile motion