Comprehensive theory, key formulas, diagrams, and memory aids for Vector Algebra.
Vector algebra gives mathematics a language for describing quantities that have both magnitude and direction, such as velocity, force, and displacement. Unlike scalars, which are described by a single number, vectors require both a magnitude and a direction for their complete specification. This chapter formalises vectors as directed line segments, develops the arithmetic of vectors, and establishes the two central products: the scalar (dot) product and the vector (cross) product.
The chapter begins with the classification of vectors, including zero, unit, and position vectors, and their representation in terms of components along the coordinate axes. It then develops vector addition, scalar multiplication, and the section formula, followed by the dot product, its geometric meaning, and orthogonality conditions. The cross product and its role in computing areas and finding vectors perpendicular to two given vectors complete the chapter.
Vector algebra is the indispensable preparation for the next chapter on three-dimensional geometry, where lines and planes are described using vectors, and for physics, where forces, velocities, and fields are all vector quantities. The chapter also introduces the fundamental idea of a vector-valued product that produces both a scalar (the dot product) and a vector (the cross product), concepts that pervade higher mathematics and mechanics.
A vector is represented by a directed line segment such as $\overrightarrow{AB}$, with initial point A and terminal point B. The magnitude of $\overrightarrow{AB}$ is denoted $|\overrightarrow{AB}|$.
Using the unit vectors $\hat{i}$, $\hat{j}$, $\hat{k}$ along the x, y, and z axes, any vector can be written as
$$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$
Its magnitude is $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$, and its direction cosines are $l = \frac{a_1}{|\vec{a}|}$, $m = \frac{a_2}{|\vec{a}|}$, $n = \frac{a_3}{|\vec{a}|}$, satisfying $l^2 + m^2 + n^2 = 1$.
If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, then
$$\vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}$$
$$k\vec{a} = ka_1\hat{i} + ka_2\hat{j} + ka_3\hat{k}$$
If P divides the line segment AB in the ratio m : n, then
$$\overrightarrow{OP} = \frac{m\vec{b} + n\vec{a}}{m + n}$$
for internal division, and the same formula with a minus sign (or with the ratio taken with sign) for external division.
The dot product of $\vec{a}$ and $\vec{b}$ is the scalar
$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta$$
where $\theta$ is the angle between the vectors. In component form:
$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$$
The cross product of $\vec{a}$ and $\vec{b}$ is the vector
$$\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin\theta \, \hat{n}$$
where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$, directed according to the right-hand rule. Its magnitude $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ equals the area of the parallelogram formed by the two vectors.
In component form:
$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}$$
The scalar triple product is
$$[\vec{a}\, \vec{b}\, \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix}$$
Its magnitude gives the volume of the parallelepiped formed by the three vectors. The three vectors are coplanar if and only if the scalar triple product is zero.
| Concept | Definition/Formula |
|---|---|
| Magnitude | |
| Unit vector | a-hat = a/ |
| Dot product | a.b = |
| Cross product | a x b = |
| Angle | cos(theta) = (a.b)/( |
| Projection of a on b | (a.b)/ |
| Property | Dot product | Cross product |
|---|---|---|
| Result | Scalar | Vector |
| Zero when | Perpendicular | Parallel |
| Commutative? | Yes | No, anti-commutative |
| a with itself | a | |
| Geometric meaning | Projection | Area of parallelogram |
| Division | Formula |
|---|---|
| Internal in ratio m:n | OP = (m b + n a)/(m + n) |
| External in ratio m:n | OP = (m b - n a)/(m - n) |
| Midpoint | (a + b)/2 |
| Centroid of triangle | (a + b + c)/3 |
graph TD
A["Vector Algebra"] --> B["Basic Vectors"]
A --> C["Addition and Scalar Multiplication"]
A --> D["Dot Product"]
A --> E["Cross Product"]
A --> F["Section Formula"]
A --> G["Scalar Triple Product"]
B --> B1["Zero, unit, position vectors"]
B --> B2["Component form: a1i + a2j + a3k"]
C --> C1["Triangle law of addition"]
D --> D1["a.b = |a||b| cos(theta)"]
D --> D2["Zero when perpendicular"]
E --> E1["a x b = |a||b| sin(theta) n-hat"]
E --> E2["Zero when parallel"]
E --> E3["Area of parallelogram"]
F --> F1["Internal and external division"]
G --> G1["a.(b x c)"]
G --> G2["Zero when coplanar"]
Vector algebra provides a compact and powerful framework for all of geometry and mechanics. The classification of vectors, component representation, and the laws of addition give the arithmetic backbone, while the dot product captures projections and perpendicularity and the cross product captures areas and perpendicular directions. The section formula connects vectors to the geometry of division of line segments, and the scalar triple product unifies the three-vector analysis through volume and coplanarity conditions. Every concept in this chapter feeds directly into the next chapter on three-dimensional geometry, where the equations of lines and planes are naturally expressed in vector form, and into physics, where the full power of vectors is required. Mastery of these algebraic skills is therefore essential for both examination success and for all subsequent work involving directed quantities.