JEE Main 2023 · 24 January · morning · Numerical answer · +4 / −0
Vectors aî+bĵ+k̂ and 2î-3ĵ+4k̂ are perpendicular to each other when 3a+2b=7, the ratio of a to b is x/2. The value of x is ____________.
Answer and solution
Answer 1
Solution
Physics sir · 4 steps
Perpendicular means the dot product is zero. That gives a second equation, so solve the pair together.
- Perpendicular condition. Two vectors are perpendicular when their dot product is zero. Multiply matching components and add them.
(a)(2) + (b)(-3) + (1)(4) = 0 ⇒ 2a - 3b = -4
- Eliminate b. Multiply 2a - 3b = -4 by 2 to get 4a - 6b = -8. Multiply 3a + 2b = 7 by 3 to get 9a + 6b = 21. Adding the two gives 13a = 13, so a = 1.
- Find b. Put a = 1 into 3a + 2b = 7: 3 + 2b = 7, so b = 2. Check: 2(1) - 3(2) + 4 = 0.
- So. a/b = 1/2 = x/2, which gives x = 1.
The common slip: 0.5. Reporting the ratio a/b = 0.5 itself gives 0.5. The question asks for x, and x/2 = 1/2 means x = 1.
The idea to keep. Whenever vectors are perpendicular, write A⃗ · B⃗ = 0 straight away. It turns geometry into one linear equation in the unknown components.
Answer from the NTA final answer key. Solution by PrepWiser.