Introduction To Vector Operations for JEE and CBSE board exams

Concepts to remember:

Vector Operations:

• Addition and subtraction of vectors: Add vectors head to tail, and subtract vectors by adding the negative of one to the other.

• Scalar and vector multiplication: Scalar multiplication multiplies a vector by a number. Vector multiplication has two forms: cross-products (for 3D vectors) and dot products (for vectors of any dimension).

• Cross product of two vectors: The cross product of two 3D vectors is a vector that is perpendicular to both of the original vectors.

• Dot product of two vectors: The dot product of two vectors is a scalar that is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them.

• Magnitude of a vector: The magnitude of a vector is the length of the vector. For 2D vectors, it is the distance between the head and the tail. For 3D vectors, it is the length of the diagonal of the rectangular box formed by the components.

• Unit vector: A unit vector is a vector with magnitude 1.

Vector Applications

• Position vector: A position vector is a vector that represents the position of a point in space relative to a fixed origin.

• Displacement vector: A displacement vector is a vector that represents the change in position of an object from one point to another.

• Velocity vector: A velocity vector is a vector that represents the rate of change of displacement with respect to time.

• Acceleration vector: An acceleration vector is a vector that represents the rate of change of velocity with respect to time.

Vector Properties:

• Scalar and vector quantities: Scalar quantities have only magnitude, while vector quantities have both magnitude and direction.

• Vector components: The components of a vector are the projections of the vector onto the coordinate axes.

• Vector algebra: Vector algebra is the branch of mathematics that deals with vectors and vector operations.