If and then. The unit vector in the direction of is . The value of is
Show answer and solution
Answer: 4
Solution
Compute P × Q using the determinant formula.
P × Q = |î ĵ k̂|
|3 √3 2|
|4 √3 2.5|
î component: (√3 × 2.5 − 2 × √3) = 2.5√3 − 2√3 = 0.5√3
ĵ component: −(3 × 2.5 − 2 × 4) = −(7.5 − 8) = 0.5
k̂ component: (3√3 − 4√3) = −√3
So P × Q = 0.5√3 î + 0.5 ĵ − √3 k̂
Factor out 0.5: P × Q = 0.5(√3 î + ĵ − 2√3 k̂)
Magnitude of (√3 î + ĵ − 2√3 k̂) = √(3 + 1 + 12) = √16 = 4
So |P × Q| = 0.5 × 4 = 2
Unit vector = (0.5/2)(√3 î + ĵ − 2√3 k̂) = (1/4)(√3 î + ĵ − 2√3 k̂)
Comparing with (1/x)(√3 î + ĵ − 2√3 k̂), we get x = 4.
P × Q = |î ĵ k̂|
|3 √3 2|
|4 √3 2.5|
î component: (√3 × 2.5 − 2 × √3) = 2.5√3 − 2√3 = 0.5√3
ĵ component: −(3 × 2.5 − 2 × 4) = −(7.5 − 8) = 0.5
k̂ component: (3√3 − 4√3) = −√3
So P × Q = 0.5√3 î + 0.5 ĵ − √3 k̂
Factor out 0.5: P × Q = 0.5(√3 î + ĵ − 2√3 k̂)
Magnitude of (√3 î + ĵ − 2√3 k̂) = √(3 + 1 + 12) = √16 = 4
So |P × Q| = 0.5 × 4 = 2
Unit vector = (0.5/2)(√3 î + ĵ − 2√3 k̂) = (1/4)(√3 î + ĵ − 2√3 k̂)
Comparing with (1/x)(√3 î + ĵ − 2√3 k̂), we get x = 4.
[JEE Main 2023 · 25 Jan · Shift 1]