Vector Algebra Notes: Class 12 Mathematics Chapter 10
Class 12 Mathematics Chapter 10 Vector Algebra covers the fundamentals of vectors, including their representation, addition, the dot and cross products, and their use in finding angles and projections between vectors.
Overview of Topics & Subtopics (Click to Jump)
1. Representation and Basic Operations
Short Note:
• A vector is represented by its magnitude and direction, and can be expressed in component form as a = xi + yj + zk, with magnitude |a| = √(x² + y² + z²).
• Vectors are added using the triangle law or parallelogram law, and scalar multiplication of a vector changes its magnitude while (for a positive scalar) preserving its direction.
2. Dot Product (Scalar Product)
Short Note:
• The dot product of two vectors a and b is defined as a · b = |a||b|cosθ, where θ is the angle between them, and it results in a scalar quantity.
• The dot product is used to determine the angle between two vectors and to test perpendicularity, since a · b = 0 if and only if the vectors are perpendicular (assuming neither is a zero vector).
3. Cross Product (Vector Product)
Short Note:
• The cross product of two vectors a and b is defined as a × b = |a||b|sinθ n̂, where n̂ is a unit vector perpendicular to both a and b, following the right-hand rule.
• The magnitude of the cross product represents the area of the parallelogram formed by the two vectors, making it useful in geometry and physics problems.
NCERT Textbook Exercise Questions
How do you find the angle between two vectors using the dot product?
Answer: Given two vectors a and b, the angle θ between them can be found by rearranging the dot product formula a · b = |a||b|cosθ to get cosθ = (a · b) / (|a||b|). Computing the dot product and the magnitudes of both vectors, and then taking the inverse cosine of this ratio, gives the required angle.
What condition indicates that two non-zero vectors are perpendicular to each other?
Answer: Two non-zero vectors a and b are perpendicular to each other if and only if their dot product equals zero, that is, a · b = 0. This follows directly from the dot product formula, since cos 90° = 0.
What does the magnitude of the cross product of two vectors represent geometrically?
Answer: The magnitude of the cross product of two vectors, |a × b|, represents the area of the parallelogram formed with the two vectors as adjacent sides. Consequently, half of this magnitude gives the area of the triangle formed by the two vectors.
Explain the difference between the dot product and the cross product of two vectors.
Answer: The dot product of two vectors results in a scalar quantity and is calculated using the cosine of the angle between them, primarily used to find angles and check perpendicularity. The cross product, on the other hand, results in a vector quantity that is perpendicular to both original vectors, calculated using the sine of the angle between them, and is primarily used to find areas and a direction perpendicular to a given plane.
