Distance from a point to a plane

3.6.11 · Maths › 3D Geometry

HUM KYA MEASURE kar rahe hain?

Plane ko aise likha jaata hai: $\pi :ax+by+cz+d=0,$ aur iska normal vector $\vec{n}=(a,b,c)$ hai, kyunki yahi woh direction hai jis taraf plane "face" karti hai.

KAISE: formula scratch se derive karo

Maano $P=(x_1,y_1,z_1)$ hamara point hai, aur $Q=(x_0,y_0,z_0)$ koi bhi aisa point hai jo plane par lie karta ho (yaani $a{x}_0+b{y}_0+c{z}_0+d=0$).

Plane par ek point kyun lete hain? Kyunki distance aslam mein vector $\overrightarrow{QP}$ ka normal par projection hai — aur projection compute karne ke liye mujhe ek tail plane par chahiye aur head $P$ par.

Step 1 — plane se point tak vector banao. $\overrightarrow{QP}=(x_1-x_0,y_1-y_0,z_1-z_0)$ Yeh step kyun? Yeh vector plane se $P$ tak jaata hai; $\vec{n}$ ke saath iska shadow exactly perpendicular distance hai.

Step 2 — unit normal par project karo. $\overrightarrow{QP}$ ka component $\hat{n}=\frac{\vec{n}}{|\vec{n}|}$ ke direction mein signed distance deta hai: $D=\overrightarrow{QP}\cdot \hat{n}=\frac{\vec{n}\cdot \overrightarrow{QP}}{|\vec{n}|}.$ Yeh step kyun? Kisi vector ka ek unit direction par projection dot product hota hai — yeh normal ke saath pointing part ki length deta hai, aur sirf yahi part matter karta hai.

Step 3 — dot product expand karo. $\vec{n}\cdot \overrightarrow{QP}=a(x_1-x_0)+b(y_1-y_0)+c(z_1-z_0)$ $=a{x}_1+b{y}_1+c{z}_1-(a{x}_0+b{y}_0+c{z}_0).$

Step 4 — use karo ki $Q$ plane par hai. Kyunki $a{x}_0+b{y}_0+c{z}_0=-d$, substitute karo: $\vec{n}\cdot \overrightarrow{QP}=a{x}_1+b{y}_1+c{z}_1+d.$ Yeh step kyun? Unknown point $Q$ magically gayab ho jaata hai — answer sirf $P$ aur plane ke coefficients par depend karta hai.

Step 5 — $|\vec{n}|$ se divide karo aur magnitude lo (distance non-negative hoti hai):

Numerator = "point ko plane equation mein daalo." Denominator = "normal ki length," jo us raw number ko ek sahi distance mein convert karti hai.

Sign batata hai aap kis side ho

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

• Equation of a plane — $(a,b,c,d)$ kahan se aate hain.
• Normal vector to a plane — $\vec{n}=(a,b,c)$ kyun.
• Projection of a vector — dot-product step ek projection hai.
• Distance between two parallel planes — seedha corollary.
• Distance from a point to a line in 3D — sister formula cross products use karta hai.
• Foot of perpendicular from a point to a plane — actual landing point nikalo.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps