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1. Chapter 10 Class 12 Vector Algebra
2. Serial order wise
3. Miscellaneous

Transcript

Misc 1 Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.Let the unit vector be 𝑎 ⃗ We know that 𝑎 ⃗ = 𝑥𝑖 ̂ + y𝑗 ̂ + z𝑘 ̂ Since the vector is in X-Y plane, there is no z– coordinate. Hence, 𝑎 ⃗ = x𝑖 ̂ + y𝑗 ̂ + 0𝒌 ̂ 𝒂 ⃗ = 𝒙𝒊 ̂ + y𝒋 ̂ Unit vector in direction of x axis is 𝑖 ̂ & in y axis is 𝑗 ̂ Given that 𝑎 ⃗ makes an angle of 30° with x-axis So, angle between 𝑎 ⃗ & 𝑖 ̂ is 30° Now, we know that, 𝑎 ⃗ . 𝑏 ⃗ = |𝑎 ⃗ ||𝑏 ⃗ | cos θ, Putting 𝑎 ⃗ = 𝑎 ⃗ , 𝑏 ⃗ = 𝑖 ̂ & θ = 30° 𝒂 ⃗ .𝒊 ̂ = |𝒂 ⃗ ||𝒊 ̂ | cos 30° 𝑎 ⃗ .𝑖 ̂ = 1 × 1 × √3/2 𝑎 ⃗ .𝑖 ̂ = √3/2 (As 𝑎 ⃗ is unit vector, |𝑎 ⃗ | = 1 & 𝑖 ̂ is a unit vector, |𝑖 ̂ | = 1) (𝑥𝑖 ̂ + y𝑗 ̂ + 0𝑘 ̂). 𝑖 ̂ = √3/2 (𝑥𝑖 ̂ + y𝑗 ̂ + 0𝑘 ̂). (1𝑖 ̂ + 0𝑗 ̂ + 0𝑘 ̂) = √3/2 𝑥.1 + y.0 + 0.0 = √3/2 𝒙=√𝟑/𝟐 Similarly, 𝑎 ⃗ makes an angle of 60° with y-axis So, angle between 𝒂 ⃗ & 𝒋 ̂ is 60° Now, we know that, 𝑎 ⃗ . 𝑏 ⃗ = |𝑎 ⃗ ||𝑏 ⃗ | cos θ, Putting 𝑎 ⃗ = 𝑎 ⃗ , 𝑏 ⃗ = 𝑗 ̂ & θ = 60° 𝒂 ⃗ .𝒋 ̂ = |𝒂 ⃗ ||𝒋 ̂ | cos 60° 𝑎 ⃗ .𝑗 ̂ = 1 × 1 × 1/2 𝑎 ⃗ .𝑗 ̂ = 1/2 (𝑥𝑖 ̂ + y𝑗 ̂ + 0𝑘 ̂). 𝑗 ̂ = 1/2 (𝑥𝑖 ̂ + y𝑗 ̂ + 0𝑘 ̂). (0𝑖 ̂ + 1𝑗 ̂ + 0𝑘 ̂) = 1/2 𝑥.0 + y.1 + 0.0 = 1/2 y = 𝟏/𝟐 Thus, 𝑎 ⃗ = 𝑥𝑖 ̂ + y𝑗 ̂ = √𝟑/𝟐 𝒊 ̂ + 𝟏/𝟐 𝒋 ̂

Miscellaneous 