Class 12 Mathematics Chapter 10: Vector Algebra
CBSE 2026–27 | Detailed NCERT-aligned study resource
Vector Algebra covers vectors and scalars, magnitude and direction, direction cosines/ratios, types of vectors, components, section formula, scalar product, vector product and their applications. The CBSE framework also includes the scalar triple product. citeturn0search12
1. Scalar and Vector
A scalar has magnitude only; a vector has magnitude and direction. Displacement, velocity and force are common vector examples.
|A|=√(a₁²+a₂²+a₃²)
2. Types of Vectors
- Zero vector: magnitude 0.
- Unit vector: magnitude 1.
- Equal vectors: same magnitude and direction.
- Parallel/collinear vectors: parallel directions or opposite directions.
- Negative vector: same magnitude, opposite direction.
3. Unit Vector and Components
The unit vector along A is A/|A|. If A=3i+4j, |A|=5 and the unit vector is (3/5)i+(4/5)j.
4. Direction Cosines and Ratios
If a vector makes angles α, β, γ with the positive axes, l=cosα, m=cosβ, n=cosγ and l²+m²+n²=1. Direction ratios are proportional to its components.
5. Addition and Scalar Multiplication
Add vectors component by component. Multiplication by a scalar changes magnitude; a negative scalar reverses direction.
6. Section Formula
If P divides the segment joining position vectors a and b internally in ratio m:n, then P=(mb+na)/(m+n).
AP:PB=m:n
7. Scalar Product
A·B=|A||B|cosθ=a₁b₁+a₂b₂+a₃b₃. It gives a scalar. For non-zero vectors, A·B=0 means perpendicular vectors.
Projection: scalar projection of A on B=(A·B)/|B|.
Example
A=i+2j+2k and B=2i−j+k. Then A·B=2−2+2=2.
8. Vector Product
A×B is perpendicular to both vectors and has magnitude |A||B|sinθ. Its direction follows the right-hand rule. The magnitude equals the area of the parallelogram formed by A and B.
|A×B| = parallelogram area
½|A×B| = triangle area
9. Scalar Triple Product
The scalar triple product is A·(B×C). Its absolute value represents the volume of the parallelepiped formed by the three vectors. If A·(B×C)=0, the vectors are coplanar.
10. Dot vs Cross Product
| Property | Dot | Cross |
|---|---|---|
| Result | Scalar | Vector |
| Formula | |A||B|cosθ | |A||B|sinθ |
| Zero condition | Perpendicular | Parallel |
| Application | Angle/projection | Area/direction |
11. Worked Area Example
Let A=i+2j and B=3i+j. In the plane, the cross-product magnitude is |1·1−2·3|=5. Therefore parallelogram area=5 square units and triangle area=5/2 square units.
12. Practice Set
- Find magnitude and unit vector of 2i−3j+6k.
- Find direction cosines of a given vector.
- Find the point dividing two position vectors in a given ratio.
- Find the angle between two vectors using dot product.
- Find the projection of one vector on another.
- Find the area of a triangle using cross product.
- Test three vectors for coplanarity using scalar triple product.
Worked Vector Example — Perpendicularity
Let A=2i−j+k and B=i+2j. Then A·B=2−2+0=0. Since the vectors are non-zero and their dot product is zero, they are perpendicular.
Common Mistakes — Chapter 10
- Confusing dot product with cross product.
- Forgetting that cross-product direction depends on order: A×B=−(B×A).
- Using direction cosines without ensuring l²+m²+n²=1.