Misc 19 - Chapter 10 Class 12 Vector Algebra

Transcript

Misc 19 If θ is the angle between any two vectors 𝑎 ⃗ and 𝑏 ⃗, then |𝑎 ⃗" ⋅" 𝑏 ⃗ | = |𝑎 ⃗" × " 𝑏 ⃗ | when θ is equal to (A) 0 (B) 𝜋/4 (C) 𝜋/2 (D) π 𝑎 ⃗" ⋅" 𝑏 ⃗ = |𝑎 ⃗ ||𝑏 ⃗ | cos⁡"θ" |𝑎 ⃗" ⋅" 𝑏 ⃗ | = ||𝑎 ⃗ ||𝑏 ⃗ |cos⁡"θ" | = |𝑎 ⃗ ||𝑏 ⃗ | cos⁡"θ" (𝑎 ⃗×𝑏 ⃗ ) = |𝑎 ⃗ ||𝑏 ⃗ | 𝑠𝑖𝑛⁡"θ" 𝑛 ̂ |𝑎 ⃗×𝑏 ⃗ |= ||𝑎 ⃗ ||𝑏 ⃗ |sin⁡〖"θ " 𝑛 ̂ 〗 | = |𝑎 ⃗ ||𝑏 ⃗ |𝑠𝑖𝑛 θ |𝒏 ̂ | = |𝑎 ⃗ ||𝑏 ⃗ |𝑠𝑖𝑛 θ Given, |𝑎 ⃗" ⋅" 𝑏 ⃗ | = |𝑎 ⃗×𝑏 ⃗ | So, |𝑎 ⃗ ||𝑏 ⃗ | cos θ = |𝑎 ⃗ ||𝑏 ⃗ | sin θ cos θ = sin θ We know that cos 𝜋/4 = sin 𝜋/4 = 1/√2 ∴ θ = 𝝅/𝟒 Therefore, the angle between the vectors 𝑎 ⃗ and 𝑏 ⃗ is 𝜋/4, So, option (B) is correct.