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3.Scalar & Vector Products
Class 12 • Mathematics • NCERT
Scalar & Vector Products
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This lesson explains the scalar (dot) product and vector (cross) product of vectors, their properties, and applications, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand scalar (dot) product of vectors.
- Understand vector (cross) product of vectors.
- Learn properties and geometric interpretation.
- Solve NCERT board-level questions.
1. Scalar (Dot) Product
The scalar product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them.
→a · →b = |a||b| cosθ
[ Image Placeholder: Angle Between Two Vectors ]
2. Properties of Scalar Product
The scalar product satisfies the following properties:
• →a · →b = →b · →a (Commutative)
• →a · (→b + →c) = →a · →b + →a · →c
• →a · →a = |a|²
• →a · (→b + →c) = →a · →b + →a · →c
• →a · →a = |a|²
[ Image Placeholder: Projection of a Vector ]
3. Angle Between Two Vectors
The angle between two non-zero vectors can be found using the scalar product.
cosθ = (→a · →b) / (|a||b|)
[ Image Placeholder: Finding Angle Using Dot Product ]
4. Vector (Cross) Product
The vector product of two vectors is a vector perpendicular to the plane containing them.
→a × →b = |a||b| sinθ n̂
[ Image Placeholder: Right-Hand Thumb Rule ]
5. Properties of Vector Product
• →a × →b = − (→b × →a)
• →a × →a = 0
• →a × (→b + →c) = →a × →b + →a × →c
• →a × →a = 0
• →a × (→b + →c) = →a × →b + →a × →c
[ Image Placeholder: Direction of Cross Product ]
6. Important NCERT Notes
• Dot product gives a scalar result
• Cross product gives a vector result
• Dot product is zero for perpendicular vectors
• Cross product is zero for parallel vectors
• Cross product gives a vector result
• Dot product is zero for perpendicular vectors
• Cross product is zero for parallel vectors
Practice Questions (NCERT)
- Define scalar product of two vectors.
- Write the formula for dot product.
- When is dot product zero?
- Define vector product.
- Write the formula for cross product.
- State one property of vector product.
- What does →a · →a represent?
- What is direction of →a × →b?
- When is cross product zero?
- Is dot product commutative?
✅ Show Answer Key
- Product giving scalar value
- |a||b|cosθ
- When vectors are perpendicular
- Product giving a vector
- |a||b|sinθ n̂
- Anti-commutative
- Square of magnitude
- Perpendicular to plane of vectors
- When vectors are parallel
- Yes
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