Coefficients normal KYUN hain? Do points lo P=(x1,y1,z1) aur Q=(x2,y2,z2) jo dono plane par hain. Unke equations subtract karo:
a(x1−x2)+b(y1−y2)+c(z1−z2)=0.
Left side exactly n⋅QP hai. Toh n⋅(any vector lying in the plane)=0 — matlab n plane ke har direction ke perpendicular hai. Yahi plane ke normal hone ki definition hai.
KAISE — dot product se. Kisi bhi do vectors ke liye,
n1⋅n2=∣n1∣∣n2∣cosθ.
Ye dot product ki definition hai (ek vector ka doosre par projection). Rearrange karne par:
Modulus KYUN? Ek plane ki equation ko −1 se multiply karne par (yaani poori equation flip karne par) uska normal flip ho jaata hai, aur cosθ se −cosθ ho jaata hai — lekin plane wahi rehti hai. ∣⋅∣ is sign ambiguity ko hatata hai aur hamesha chota wedge angle deta hai.
Q: Do planes ke normals ⟨1,0,0⟩ aur ⟨0,0,1⟩ hain. Compute karne se pehle angle forecast karo.
Verify: Dot =0, toh θ=90∘. Ye planes hain yz-plane aur xy-plane — ye y-axis ke saath right angle par milti hain. ✔
Line aur plane ke beech ka angle kaunsa ratio use karta hai?
sinθ=∣b∣∣n∣∣b⋅n∣ (note: sine, cosine nahi).
x+y+z=1 aur −x−y+z=2 ke beech ka angle?
cos−1(1/3)≈70.5∘.
Recall Feynman: ek 12-saal ke bacche ko explain karo
Socho do flat cardboard sheets ek doosre ke saath tiki hain, "V" bana rahi hain. Ye batane ke liye ki V kitna khula hai, tum cardboard ko touch nahi karte — har sheet se seedha ek pencil bahar nikaalo aur dono pencils ke beech ka angle measure karo. Wahi pencils "normals" hain. 2x−y+2z=5 jaisi plane ke liye, pencil bas wahi teen numbers hain jo aage likhe hain: (2,−1,2). Matching numbers multiply karo, add karo, lengths se divide karo — aur calculator ka cosine button angle bata dega. Hum answer hamesha positive rakhte hain taaki chota, samajh mein aane wala angle mile.
