Shortest Distance Using Cross Product

 

❓ Concept

JEE 3D geometry mein jab shortest distance diya ho
aur α parameter direction ratios mein aa jaaye —
samajh lo cross product ka real test start ho gaya hai 🔥

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🖼️ Concept Image

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✍️ Short Explanation

Shortest distance problems formula-based hote hain,
lekin jab parameter (α) involve ho, toh goal hota hai:

👉 SD ko α ke function mein convert karna
👉 Given distance se equation banana
👉 α ke possible values nikaalna (ya unka sum)

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🔹 Step 1 — Lines in Symmetric Form

Agar line di ho:

$$\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$$

Toh:

• Direction vector

$$\vec d_1=(a_1,b_1,c_1)$$

• Point on line

$$A(x_1,y_1,z_1)$$

Similarly, second line ke liye:

$$\vec d_2=(a_2,b_2,c_2)$$

and point $B(x_2,y_2,z_2)$.

📌 Agar α direction ratios mein hai,
toh $\vec d_2$ α pe depend karega.

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🔹 Step 2 — Shortest Distance Formula (CORE 🔥)**

For skew lines, shortest distance:

$$\text{SD} = \frac{|(\vec B-\vec A)\cdot(\vec d_1\times\vec d_2)|} {|\vec d_1\times\vec d_2|}$$

⚠️ Important:

• Yeh formula sirf skew lines ke liye valid hai

• Parallel ya intersecting lines mein SD = 0 ya different logic

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🔹 Step 3 — Role of Parameter (α)

Jab α aata hai:

• ${\vec{d}}_2=f(\alpha )$

• $\vec d_1\times\vec d_2 = g(\alpha)$
  

• Numerator & denominator dono α pe depend karte hain

👉 Isliye:

$$\text{SD} = \text{function of } \alpha$$

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🔹 Step 4 — Use the Given Shortest Distance

Agar question bole:

$$\text{SD} = k$$

Toh:

$$\frac{|(\vec B-\vec A)\cdot(\vec d_1\times\vec d_2)|} {|\vec d_1\times\vec d_2|} = k$$

Next steps:
1️⃣ Square both sides (modulus & root hat jaata hai)
2️⃣ Simplify
3️⃣ Quadratic equation in α mil jaati hai

Usually:

$$A\alpha^2+B\alpha+C=0$$

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🔹 Step 5 — Final JEE Insight (SMART MOVE 😎)**

Agar question pooche:

“Sum of all possible values of α”

Toh:

• Roots = $\alpha_1,\alpha_2$

• Sum of roots:

$$\alpha_1+\alpha_2 = -\frac{B}{A}$$

👉 Individual α nikaalne ki zarurat hi nahi!

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✅ Final Takeaway

🧠 Shortest Distance (α-based) Golden Rules

• Direction vectors pehle identify karo

• Cross product nikaalo

• SD formula lagao

• Given value se equation banao

• Sum of α = −B/A

Is trick se parameter wale 3D questions JEE mein lightning-fast ho jaate hain ⚡