Class 12 Mathematics PRODUCT OF VECTORS Notes

Class 12 Mathematics PRODUCT OF VECTORS Notes
Product of Vectors Notes | Nepal eNotes

Unit 12

Vectors

Class 12 Mathematics

Product of Vectors

Class 12 Mathematics – Product of Vectors Notes PDF

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Contents / Quick Navigation

1. Product of Vectors2. Scalar Product3. Angle Between Vectors 4. Properties of Scalar Product5. Scalar Product Applications 6. Vector Product7. Properties of Vector Product 8. Geometric Applications9. Worked Examples 10. Common Mistakes11. Exam Questions12. One-Minute Revision

NEB / CDC Focus

This unit covers the scalar (dot) product and vector (cross) product of two vectors, their geometric interpretation and properties, the angle between vectors, and applications of these products in trigonometry and geometry.

1. Product of Vectors

Multiplying vectors is not the same as multiplying ordinary numbers. Two principal products are used:

ProductNotationResultMain use
Scalar / dot producta·bScalarAngles, projections, perpendicularity
Vector / cross producta×bVectorArea, perpendicular direction, parallelism
Two Products of Vectors Dot Product a·bProduces a scalarmeasures directional alignment Cross Product a×bProduces a vectorperpendicular to both vectors
Figure 1: Dot and cross products have different outputs and geometric meanings.

2. Scalar Product of Two Vectors

Definition

If vectors a and b make angle θ, their scalar product is

a·b = |a||b|cosθ.

For components a=a₁i+a₂j+a₃k and b=b₁i+b₂j+b₃k:

a·b=a₁b₁+a₂b₂+a₃b₃.

Basic unit-vector products:

i·i=j·j=k·k=1i·j=j·k=k·i=0.
Geometric Meaning of a·b θbdirection of a projection component = |b|cosθ
Figure 2: The dot product is magnitude of one vector times the projection of the other along it.

3. Angle Between Two Vectors

cosθ=(a·b)/(|a||b|).

Thus

θ=cos⁻¹[(a·b)/(|a||b|)].
Perpendicular Vectors

For non-zero vectors, a·b=0 ⇔ a⊥b.

Worked Example

Find the angle between a=(1,1,0) and b=(1,0,1).

a·b=1, |a|=√2, |b|=√2 ⇒ cosθ=1/2 ⇒ θ=60°.

4. Properties of Scalar Product

PropertyFormula
Commutativea·b=b·a
Distributivea·(b+c)=a·b+a·c
Scalar multiplication(λa)·b=λ(a·b)
Self producta·a=|a|²
Orthogonalitya·b=0 for non-zero perpendicular vectors
Useful Identities
|a+b|²=|a|²+|b|²+2a·b|a−b|²=|a|²+|b|²−2a·b (a+b)·(a−b)=|a|²−|b|²

5. Applications of Scalar Product

5.1 Projection

Scalar projection of a on b:

comp_b a=(a·b)/|b|.

Vector projection:

proj_b a=[(a·b)/|b|²]b.

5.2 Proving Trigonometric Relations

Taking dot products of unit vectors with known included angles can reproduce cosine rules and related identities.

Cosine Rule from Vectors

If c=a−b, then

|c|²=|a|²+|b|²−2|a||b|cosθ.

Vector Projection aproj_b ab direction
Figure 3: Projection extracts the component of one vector along another.

6. Vector Product of Two Vectors

Definition

The vector product of vectors a,b is a vector perpendicular to both:

a×b=|a||b|sinθ n̂

where n̂ is the unit normal determined by the right-hand rule.

For components:

a×b = | i   j   k ; a₁   a₂   a₃ ; b₁   b₂   b₃ |
=(a₂b₃−a₃b₂)i −(a₁b₃−a₃b₁)j +(a₁b₂−a₂b₁)k.

Unit-vector products:

i×j=k, j×k=i, k×i=jj×i=−k, k×j=−i, i×k=−j.
Cross Product Direction aba×bθ
Figure 4: a×b is perpendicular to the plane containing a and b.

7. Properties of Vector Product

PropertyFormula
Anti-commutativea×b=−(b×a)
Distributivea×(b+c)=a×b+a×c
Self producta×a=0
Parallel conditiona×b=0 for non-zero parallel vectors
Magnitude|a×b|=|a||b|sinθ
Order Matters

The cross product is not commutative. Reversing the order reverses the direction.

8. Geometric Applications of Vector Product

8.1 Area of a Parallelogram

Area=|a×b|.

8.2 Area of a Triangle

Area=(1/2)|a×b|.

8.3 Unit Normal

n̂=(a×b)/|a×b|.
Area from a Cross Product height=|b|sinθ|a| Area=base×height=|a||b|sinθ=|a×b|
Figure 5: The magnitude of the cross product is the parallelogram area.
Worked Example

Find the area of the triangle formed by a=(1,0,0), b=(0,2,0).

a×b=(0,0,2), |a×b|=2 ⇒ triangle area=1.

9. Worked Examples

Example 1: Dot Product

For a=(2,−1,3), b=(1,4,−2):

a·b=2−4−6=−8.

Example 2: Perpendicularity

Check (1,2,−1) and (2,−1,0).

1·2+2(−1)+(−1)0=0 ⇒ perpendicular.

Example 3: Cross Product

Find (1,2,3)×(2,0,1).

=(2,5,−4).

Example 4: Parallel Test

(1,2,3)×(2,4,6)=0, hence the vectors are parallel.

Example 5: Projection

Projection of a=(3,4,0) on b=(1,0,0) has scalar value 3.

Quick Decision: Dot or Cross? Need angle / projection / ⟂?Use a·b Need area / normal / ∥?Use a×b
Figure 6: A quick method-selection guide.

10. Common Mistakes

Dot vs cross

Dot product gives a scalar; cross product gives a vector.

Wrong cross order

a×b=−b×a.

Missing magnitudes

In a·b=|a||b|cosθ, both magnitudes are essential.

Wrong cross j-sign

When expanding the determinant, the middle component carries a minus sign.

Area factor

Parallelogram area is |a×b|; triangle area is half.

Zero vector caveat

Use dot/cross parallel-perpendicular tests carefully when a vector is zero.

11. Formula Sheet

a·b=|a||b|cosθ=a₁b₁+a₂b₂+a₃b₃ cosθ=(a·b)/(|a||b|) proj_b a=[(a·b)/|b|²]b a×b=|a||b|sinθ n̂ Area(parallelogram)=|a×b| Area(triangle)=½|a×b|

12. Important Exam Questions

Short

  1. Define scalar product and vector product.
  2. State geometric interpretations of dot and cross products.
  3. State conditions for perpendicular and parallel non-zero vectors.
  4. Write unit-vector dot and cross product tables.

Long / Numerical

  1. Find angle between two vectors using dot product.
  2. Find scalar/vector projection.
  3. Prove a vector identity using dot-product properties.
  4. Find a cross product from components.
  5. Find area of triangle/parallelogram from vectors.
  6. Find a unit vector perpendicular to two given vectors.

13. One-Minute Revision

  • Dot product produces a scalar.
  • Cross product produces a perpendicular vector.
  • a·b=|a||b|cosθ.
  • For non-zero perpendicular vectors, a·b=0.
  • a×b=|a||b|sinθ n̂.
  • For non-zero parallel vectors, a×b=0.
  • Cross product changes sign when order reverses.
  • |a×b| is parallelogram area.
  • Half of it is triangle area.
  • Normalize a×b to obtain a unit normal.

14. Diagram Practice

  • Dot vs cross product overview.
  • Projection diagram.
  • Angle between two vectors.
  • Right-hand-rule cross-product direction.
  • Parallelogram area.
  • Dot-or-cross method decision diagram.

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
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