Vector Algebra - Addition of vectors, multiplication of a vector by a scalar

Position Vector: The position vector of a point $P(x, y, z)$ with respect to the origin $O(0, 0, 0)$ is given by $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$. Visually, this is represented as a directed arrow starting at the origin and ending at the point $P$ in three-dimensional space.

Direction Cosines and Ratios: The coordinates $x, y, z$ are the direction ratios of the position vector. The direction cosines $\cos \alpha, \cos \beta, \cos \gamma$ represent the angles the vector makes with the $x, y, \text{and } z$ axes respectively, visually defining the vector's orientation relative to the coordinate frame.

Triangle Law of Vector Addition: If two vectors are represented by two sides of a triangle taken in order (head-to-tail), their sum is represented by the third side taken in the reverse order. Geometrically, this completes the triangle where the resultant vector acts as a direct path from the start of the first vector to the end of the second.

Parallelogram Law of Vector Addition: If two vectors $\vec{a}$ and $\vec{b}$ are represented by the adjacent sides of a parallelogram, their sum $\vec{a} + \vec{b}$ is represented by the diagonal passing through their common initial point. This visualizes how two concurrent forces or velocities combine into a single resultant.

Multiplication of a Vector by a Scalar: Multiplying a vector $\vec{a}$ by a scalar $\lambda$ results in a vector $\lambda\vec{a}$. Visually, the vector is scaled (stretched or shrunk) by a factor of $|\lambda|$. If $\lambda$ is positive, the direction remains the same; if $\lambda$ is negative, the direction is exactly reversed.

Unit Vector: A vector with a magnitude of $1$ is called a unit vector, denoted by $\hat{a}$. It is calculated by dividing a vector by its own magnitude. Geometrically, it represents a 'pure direction' extending one unit length from the origin along the line of the original vector.

Components of a Vector: Any vector can be expressed as $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, where $a_1, a_2, a_3$ are scalar components. Visually, these represent the projections of the vector onto the three mutually perpendicular coordinate axes, forming the edges of a rectangular box where the vector is the main diagonal.