Vector Algebra

Basic Definitions  

- Vector Representation: A vector is shown as a directed line segment, with magnitude equal to its length and direction indicated by the arrow.  

- Scalar Quantity: Has only magnitude, no direction.  

- Vector Quantity: Has both magnitude and direction.  

- Zero Vector: A vector with zero magnitude and no definite direction.  

- Unit Vector: A vector of unit length pointing in the direction of another vector.  

- Equal Vectors: Vectors with the same magnitude and direction.  

- Collinear Vectors: Vectors parallel to the same line.  

- Like/Unlike Vectors: Collinear vectors in the same direction are like; in opposite directions are unlike.  

- Coplanar Vectors: Vectors lying in the same plane.  

- Position Vector: A vector drawn from the origin to a point, representing its location in space.  

Vector Operations  

- Negative of a Vector: A vector pointing in the opposite direction.  

- Coinitial Vectors: Vectors starting from the same point.  

- Coterminal Vectors: Vectors ending at the same point.  

Addition of Vectors  

- Triangle Law: Joining vectors head‑to‑tail gives a resultant along the third side of the triangle.  

- Parallelogram Law: Two vectors form adjacent sides of a parallelogram; the diagonal represents their sum.  

- Properties: Vector addition is commutative and associative, has a zero element, and each vector has an additive inverse.  

 Subtraction of Vectors  

- Defined as adding the negative of a vector.  

- Unlike addition, subtraction is neither commutative nor associative.  

Scalar Multiplication  

- Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative, which reverses direction).  

- Follows distributive and associative properties with scalars.  

Dot Product (Scalar Product)  

- Represents the product of magnitudes and the cosine of the angle between vectors.  

- If the dot product is zero, the vectors are perpendicular.  

- Used to find projections of one vector along another.  

Cross Product (Vector Product)  

- Represents the product of magnitudes and the sine of the angle between vectors, directed perpendicular to both.  

- Zero if vectors are parallel.  

- Used to calculate areas of geometric figures like triangles.  

Scalar Triple Product  

- Involves three vectors; gives a scalar value.  

- Zero if the vectors are coplanar.  

- Represents the volume of 3D figures like parallelepipeds and tetrahedra.  

Vector Triple Product  

- Involves three vectors; results in a vector lying in the plane of two of them.  

- Shows relationships between dot and cross products.  

Distance Between Lines  

- For parallel lines, distance is constant.  

- For skew lines (non‑parallel, non‑intersecting), the shortest distance is defined using vector methods.  

Reciprocal System of Vectors  

- A special set of vectors constructed from three non‑coplanar vectors.  

- They satisfy unique properties of dot products and are useful in expressing other vectors in terms of the system.