Vector Algebra – Class 12 NCERT Summary

📘 Vector Algebra – Class 12 NCERT

Vector Algebra is an important chapter in Class 12 Mathematics. It deals with quantities having both magnitude and direction and is widely used in physics, engineering, and 3D geometry ⚙️📐.

🔹 1. Scalars and Vectors

📌 Scalars

Quantities having only magnitude are called scalars.

Examples: Mass, time, temperature, speed 🧊⏱️

📌 Vectors

Quantities having both magnitude and direction are called vectors.

Examples: Displacement, velocity, force 🚀➡️

🔹 2. Representation of Vectors

Vectors are represented geometrically by directed line segments and algebraically using unit vectors î, ĵ, k̂.

Example 🧮:
A vector a = 3î + 4ĵ represents a vector in the xy-plane.

🔹 3. Magnitude of a Vector

The magnitude (length) of a vector a = aî + bĵ + ck̂ is given by:

|a| = √(a² + b² + c²)

Example 📏:
For a = 3î + 4ĵ
|a| = √(3² + 4²) = 5

🔹 4. Direction Cosines and Direction Ratios

Direction cosines are the cosines of the angles a vector makes with the x, y, and z axes.

Example 📐:
For vector a = 2î + 2ĵ + îk̂
Direction ratios = 2, 2, 1

🔹 5. Types of Vectors

🔹 Zero vector
🔹 Unit vector
🔹 Equal vectors
🔹 Negative vectors
🔹 Collinear vectors

Example ⭐:
Unit vector along a = a / |a|

🔹 6. Addition of Vectors

Vectors can be added using the triangle law, parallelogram law, or algebraically.

Example ➕:
a = î + 2ĵ
b = 3î + ĵ
a + b = 4î + 3ĵ

🔹 7. Scalar (Dot) Product

The scalar product of two vectors a and b is defined as:

a · b = |a||b|cosθ

Example 🔵:
î · ĵ = 0
î · î = 1

📌 Applications of Dot Product

Finding angle between vectors 📐
Checking perpendicularity ⟂
Work done by a force 💪

🔹 8. Vector (Cross) Product

The vector product of two vectors a and b is defined as:

a × b = |a||b|sinθ n̂

Example 🔁:
î × ĵ = k̂
ĵ × î = −k̂

📌 Applications of Cross Product

Finding area of parallelogram 🟦
Finding area of triangle 🔺
Determining perpendicular direction

🔹 9. Scalar Triple Product

The scalar triple product a · (b × c) gives the volume of a parallelepiped.

Example 📦:
If a · (b × c) = 0, then vectors are coplanar.

🔹 10. Important Points to Remember

Vectors have both magnitude and direction ➡️
Dot product gives a scalar 🔢
Cross product gives a vector 🔄
î, ĵ, k̂ form a right-handed system ✋

✨ Conclusion

Vector Algebra builds a strong foundation for 3D geometry, physics, and engineering. Mastery of this chapter is crucial for competitive exams and higher studies 🎓🚀.

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Vector Algebra Worksheet Mathematics Class 12 PDF