📐 Angle Between Lines and Planes — A Complete Vector Geometry Guide

Author: Rishabh Kumar (IIT + ISI | Global Math Mentor)**
Published: October 2025
Category: Vectors | 3D Geometry

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🔹 Introduction

In 3D geometry, lines and planes can be parallel, perpendicular, or inclined.
The angle between them helps us understand spatial orientation — vital in geometry, mechanics, and vector analysis.

We’ll learn to find:
1️⃣ The angle between two lines
2️⃣ The angle between two planes
3️⃣ The angle between a line and a plane

Every angle in 3D can be derived from dot products — the algebraic expression of geometry.

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⚡️ 1️⃣ Angle Between Two Lines

Let two lines be: x−x1l1=y−y1m1=z−z1n1\frac{x – x_1}{l_1} = \frac{y – y_1}{m_1} = \frac{z – z_1}{n_1}l1​x−x1​​=m1​y−y1​​=n1​z−z1​​

and x−x2l2=y−y2m2=z−z2n2\frac{x – x_2}{l_2} = \frac{y – y_2}{m_2} = \frac{z – z_2}{n_2}l2​x−x2​​=m2​y−y2​​=n2​z−z2​​

Direction vectors: a1=(l1,m1,n1),a2=(l2,m2,n2)\mathbf{a_1} = (l_1, m_1, n_1), \quad \mathbf{a_2} = (l_2, m_2, n_2)a1​=(l1​,m1​,n1​),a2​=(l2​,m2​,n2​)

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🔹 Formula

cos⁡θ=a1⋅a2∣a1∣∣a2∣\boxed{\cos\theta = \frac{\mathbf{a_1}\cdot\mathbf{a_2}}{|\mathbf{a_1}||\mathbf{a_2}|}}cosθ=∣a1​∣∣a2​∣a1​⋅a2​​​

where a1⋅a2=l1l2+m1m2+n1n2\mathbf{a_1}\cdot\mathbf{a_2} = l_1l_2 + m_1m_2 + n_1n_2a1​⋅a2​=l1​l2​+m1​m2​+n1​n2​ ∣a1∣=l12+m12+n12|\mathbf{a_1}| = \sqrt{l_1^2 + m_1^2 + n_1^2}∣a1​∣=l12​+m12​+n12​​

✅ Use acute angle (0° ≤ θ ≤ 90°) for geometry problems.

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🔹 Example 1 — Angle Between Two Lines

Find the angle between x2=y3=z6\frac{x}{2} = \frac{y}{3} = \frac{z}{6}2x​=3y​=6z​

and x−11=y−1=z+22\frac{x – 1}{1} = \frac{y}{-1} = \frac{z + 2}{2}1x−1​=−1y​=2z+2​ a1=(2,3,6),a2=(1,−1,2)\mathbf{a_1} = (2, 3, 6), \quad \mathbf{a_2} = (1, -1, 2)a1​=(2,3,6),a2​=(1,−1,2) a1⋅a2=2(1)+3(−1)+6(2)=11\mathbf{a_1}\cdot\mathbf{a_2} = 2(1) + 3(-1) + 6(2) = 11a1​⋅a2​=2(1)+3(−1)+6(2)=11 ∣a1∣=22+32+62=7,∣a2∣=1+1+4=6|\mathbf{a_1}| = \sqrt{2^2 + 3^2 + 6^2} = 7, \quad |\mathbf{a_2}| = \sqrt{1 + 1 + 4} = \sqrt{6}∣a1​∣=22+32+62​=7,∣a2​∣=1+1+4​=6​ cos⁡θ=1176⇒θ=cos⁡−1(1176)\cos\theta = \frac{11}{7\sqrt{6}} \Rightarrow \theta = \cos^{-1}\left(\frac{11}{7\sqrt{6}}\right)cosθ=76​11​⇒θ=cos−1(76​11​)

✅ Angle ≈ 22.8°

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🔹 Special Cases

Condition	Meaning
a1⋅a2=0\mathbf{a_1}\cdot\mathbf{a_2} = 0a1​⋅a2​=0	Lines are perpendicular
l1l2=m1m2=n1n2\frac{l_1}{l_2} = \frac{m_1}{m_2} = \frac{n_1}{n_2}l2​l1​​=m2​m1​​=n2​n1​​	Lines are parallel

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🧭 2️⃣ Angle Between Two Planes

Planes: a1x+b1y+c1z+d1=0a_1x + b_1y + c_1z + d_1 = 0a1​x+b1​y+c1​z+d1​=0 a2x+b2y+c2z+d2=0a_2x + b_2y + c_2z + d_2 = 0a2​x+b2​y+c2​z+d2​=0

Normal vectors: n1=(a1,b1,c1),n2=(a2,b2,c2)\mathbf{n_1} = (a_1, b_1, c_1), \quad \mathbf{n_2} = (a_2, b_2, c_2)n1​=(a1​,b1​,c1​),n2​=(a2​,b2​,c2​)

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🔹 Formula

cos⁡θ=n1⋅n2∣n1∣∣n2∣\boxed{\cos\theta = \frac{\mathbf{n_1}\cdot\mathbf{n_2}}{|\mathbf{n_1}||\mathbf{n_2}|}}cosθ=∣n1​∣∣n2​∣n1​⋅n2​​​ n1⋅n2=a1a2+b1b2+c1c2\mathbf{n_1}\cdot\mathbf{n_2} = a_1a_2 + b_1b_2 + c_1c_2n1​⋅n2​=a1​a2​+b1​b2​+c1​c2​

✅ This gives the acute angle between the planes.

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🔹 Example 2 — Angle Between Two Planes

Find the angle between: 2x+y−2z+5=0andx−y+2z−3=02x + y – 2z + 5 = 0 \quad \text{and} \quad x – y + 2z – 3 = 02x+y−2z+5=0andx−y+2z−3=0 n1=(2,1,−2),n2=(1,−1,2)\mathbf{n_1} = (2, 1, -2), \quad \mathbf{n_2} = (1, -1, 2)n1​=(2,1,−2),n2​=(1,−1,2) n1⋅n2=2(1)+1(−1)+(−2)(2)=−3\mathbf{n_1}\cdot\mathbf{n_2} = 2(1) + 1(-1) + (-2)(2) = -3n1​⋅n2​=2(1)+1(−1)+(−2)(2)=−3 ∣n1∣=3,∣n2∣=6|\mathbf{n_1}| = 3, \quad |\mathbf{n_2}| = \sqrt{6}∣n1​∣=3,∣n2​∣=6​ cos⁡θ=∣−3∣36=16\cos\theta = \frac{|-3|}{3\sqrt{6}} = \frac{1}{\sqrt{6}}cosθ=36​∣−3∣​=6​1​

✅ Angle ≈ 65.9°

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🔹 Relation to Line of Intersection

The line of intersection of two planes lies within both planes and makes an angle equal to θ with either plane’s normal.

Planes meet along a line whose direction vector = n1×n2\mathbf{n_1} \times \mathbf{n_2}n1​×n2​.

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🎯 3️⃣ Angle Between a Line and a Plane

Let a line have direction vector a=(l,m,n)\mathbf{a} = (l, m, n)a=(l,m,n),
and a plane have normal vector n=(a,b,c)\mathbf{n} = (a, b, c)n=(a,b,c).

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🔹 Formula

sin⁡θ=∣a⋅n∣∣a∣∣n∣\boxed{\sin\theta = \frac{|\mathbf{a}\cdot\mathbf{n}|}{|\mathbf{a}||\mathbf{n}|}}sinθ=∣a∣∣n∣∣a⋅n∣​​

where θ = angle between the line and the plane.

✅ (The angle between line and normal = 90° − θ, hence sine instead of cosine.)

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🔹 Example 3 — Line and Plane

Find the angle between the line x−22=y+13=z−46\frac{x – 2}{2} = \frac{y + 1}{3} = \frac{z – 4}{6}2x−2​=3y+1​=6z−4​

and the plane 2x−y+2z+3=02x – y + 2z + 3 = 02x−y+2z+3=0 a=(2,3,6),n=(2,−1,2)\mathbf{a} = (2, 3, 6), \quad \mathbf{n} = (2, -1, 2)a=(2,3,6),n=(2,−1,2) a⋅n=2(2)+3(−1)+6(2)=13\mathbf{a}\cdot\mathbf{n} = 2(2) + 3(-1) + 6(2) = 13a⋅n=2(2)+3(−1)+6(2)=13 ∣a∣=7,∣n∣=3|\mathbf{a}| = 7, \quad |\mathbf{n}| = 3∣a∣=7,∣n∣=3 sin⁡θ=1321\sin\theta = \frac{13}{21}sinθ=2113​

✅ Angle = θ = sin⁻¹(13/21) ≈ 38.4°

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🔹 Special Cases

Condition	Relationship
a⋅n=0\mathbf{a}\cdot\mathbf{n} = 0a⋅n=0	Line lies in plane
a∥n\mathbf{a} \parallel \mathbf{n}a∥n	Line perpendicular to plane

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🧩 4️⃣ Summary Table

Relationship	Formula	Type	Range
Between two lines	( \cos\theta = \frac{\mathbf{a_1}\cdot\mathbf{a_2}}{	\mathbf{a_1}
Between two planes	( \cos\theta = \frac{\mathbf{n_1}\cdot\mathbf{n_2}}{	\mathbf{n_1}
Between line & plane	( \sin\theta = \frac{	\mathbf{a}\cdot\mathbf{n}	}{

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🔹 Common Mistakes

1. ❌ Mixing up sine vs cosine in line–plane formula.
2. ❌ Using wrong vectors (direction vs normal).
3. ❌ Forgetting absolute value for angle magnitude.
4. ❌ Calculating obtuse instead of acute angle.

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🌟 Why It Matters

Understanding angles in 3D is fundamental for:

• Geometry of space & mechanics,
• Computer graphics & modeling,
• IB/A Level/STEP problems on direction and orientation,
• Analytic geometry derivations.

Algebraic dot products reveal spatial relationships geometrically.

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In the Mathematics Elevate Mentorship Program, you’ll:
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