Vector Algebra - Vectors and scalars, magnitude and direction of a vector

Scalars and Vectors: A scalar quantity has only magnitude (e.g., distance, mass, time). A vector quantity has both magnitude and direction (e.g., displacement, velocity, force). Visually, a vector is represented by a directed line segment where the length of the segment denotes the magnitude and an arrowhead indicates the direction.

Position Vector: For a point $P(x, y, z)$ in a 3D coordinate system, the vector $\vec{OP}$ with the origin $O(0, 0, 0)$ as the initial point and $P$ as the terminal point is called the position vector of $P$. It is visually represented as an arrow pointing from the center of the axes to the specific point in space.

Magnitude of a Vector: The magnitude (or length) of a vector $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$ is denoted by $|\vec{a}|$. It represents the distance between the initial and terminal points. In a geometric sense, it is the actual length of the arrow representing the vector, calculated using the Pythagorean theorem in three dimensions.

Direction Cosines and Ratios: The angles $\alpha, \beta,$ and $\gamma$ made by a vector with the positive directions of the $x, y,$ and $z$ axes respectively are called direction angles. Their cosines, $l = \cos\alpha, m = \cos\beta, n = \cos\gamma$, are the direction cosines. Direction ratios are any real numbers $a, b, c$ such that $l/a = m/b = n/c$. Visually, these define the orientation of the vector in 3D space relative to the coordinate axes.

Unit Vector: A vector whose magnitude is exactly $1$ unit is called a unit vector. It is denoted by adding a 'hat' symbol (e.g., $\hat{a}$). A unit vector represents pure direction without scaling the magnitude. Geometrically, it is a vector that lies on the unit sphere centered at the origin.

Types of Vectors: (i) Zero Vector: Initial and terminal points coincide (magnitude 0). (ii) Co-initial Vectors: Vectors having the same initial point. (iii) Collinear/Parallel Vectors: Vectors parallel to the same line, regardless of their magnitudes or directions. (iv) Equal Vectors: Vectors that have the same magnitude and direction, even if they have different initial points (they are translations of each other).

Vector Joining Two Points: If we have two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, the vector $\vec{P_1P_2}$ is found by subtracting the coordinates of the initial point from the terminal point. Visually, this creates a displacement vector representing the path from $P_1$ to $P_2$.