**What are Vectors?**

Vectors are mathematical quantities that have both magnitude and direction. They are used to represent various physical quantities such as force, velocity, displacement, and acceleration. Vectors are crucial in the field of mathematics and physics as they help describe and analyze the motion and forces acting on objects.

**Vector Algebra**

Vector algebra deals with the operations and properties of vectors. It enables us to perform mathematical calculations involving vectors, such as addition, subtraction, scalar multiplication, dot product, cross product, and finding the magnitude and direction of vectors.

**1. Addition and Subtraction of Vectors:**
- Vectors can be added or subtracted by combining their corresponding components. The result is a new vector with the sum or difference of the magnitudes and the direction determined by the angle between the vectors.

**2. Scalar Multiplication:**
- Scalar multiplication involves multiplying a vector by a scalar (a real number). The magnitude of the vector is multiplied by the scalar, and the direction remains unchanged.

**3. Dot Product:**
- The dot product of two vectors is a scalar quantity obtained by multiplying the magnitudes of the vectors and the cosine of the angle between them.

**4. Cross Product:**
- The cross product of two vectors is a vector quantity that is perpendicular to both vectors. Its magnitude is the product of the magnitudes of the vectors and the sine of the angle between them.

**5. Magnitude and Direction:**
- The magnitude of a vector represents its length or size. It can be found using the Pythagorean theorem.
- The direction of a vector is given by the angle it makes with a reference axis, usually the positive x-axis.

In conclusion, vectors are mathematical quantities that possess both magnitude and direction. Vector algebra allows us to manipulate and perform calculations with vectors, including addition, subtraction, scalar multiplication, dot product, cross product, and finding the magnitude and direction of vectors. Understanding vector algebra is essential in various fields of study, particularly in physics and mathematics.

NCERT