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2.Vector Operations
Class 12 • Mathematics • NCERT
Vector Operations
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This lesson explains basic operations on vectors such as addition, subtraction, and multiplication by a scalar, as prescribed in NCERT Class 12.
Lesson Objectives
- Understand addition and subtraction of vectors.
- Apply triangle and parallelogram laws.
- Learn scalar multiplication of vectors.
- Solve NCERT board-level questions.
1. Addition of Vectors
The addition of two vectors results in a new vector called the resultant vector.
If →a and →b are two vectors, then their sum is written as →a + →b.
[ Image Placeholder: Vector Addition ]
2. Triangle Law of Vector Addition
According to the triangle law, if two vectors are represented by two sides of a triangle taken in order, then the third side represents their sum.
→AB + →BC = →AC
[ Image Placeholder: Triangle Law of Vector Addition ]
3. Parallelogram Law of Vector Addition
If two vectors are represented by two adjacent sides of a parallelogram, then the diagonal through the common point represents their resultant.
Resultant = diagonal of the parallelogram
[ Image Placeholder: Parallelogram Law ]
4. Subtraction of Vectors
Subtraction of vectors is defined as the addition of the negative of a vector.
→a − →b = →a + (−→b)
[ Image Placeholder: Vector Subtraction ]
5. Scalar Multiplication of Vectors
Multiplying a vector by a scalar changes its magnitude and may reverse its direction.
If k is a scalar and →a is a vector, then k→a is a vector.
[ Image Placeholder: Scalar Multiplication ]
6. Important NCERT Notes
• Vector addition is commutative and associative
• Vector subtraction is not commutative
• Scalar multiplication affects magnitude and direction
• Diagrams are important in board exams
• Vector subtraction is not commutative
• Scalar multiplication affects magnitude and direction
• Diagrams are important in board exams
Practice Questions (NCERT)
- Define vector addition.
- State the triangle law of vector addition.
- What does the diagonal of parallelogram represent?
- How is vector subtraction defined?
- What is scalar multiplication?
- Is vector addition commutative?
- Does scalar multiplication change direction?
- Write →a − →b in another form.
- Why diagrams are important in vector problems?
- Is vector subtraction associative?
✅ Show Answer Key
- Addition of vectors to get resultant
- Third side represents sum
- Resultant vector
- Addition of negative vector
- Multiplying vector by scalar
- Yes
- Yes, if scalar is negative
- →a + (−→b)
- To understand direction and magnitude
- No
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