NCERT Solutions for Class 12 Mathematics Chapter 10 Vector Algebra

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.

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).

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.