Vector Algebra — Study Notes

1. Introduction

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.

2. Basic Concepts

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 $|\overrightarrow{AB}|$.

Types of Vectors

• Zero vector: A vector with zero magnitude, denoted $\vec{0}$.
• Unit vector: A vector with magnitude 1, denoted $\hat{a}$ in the direction of a: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$.
• Position vector: The vector of a point P with respect to an origin O, denoted $\overrightarrow{OP}$.
• Equal vectors: Two vectors with the same magnitude and the same direction.
• Collinear (parallel) vectors: Vectors with the same or opposite directions, i.e., one is a scalar multiple of the other.
• Negative of a vector: $\vec{a}$ and $-\vec{a}$ have the same magnitude but opposite directions.

Components Along the Axes

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 $l^2 + m^2 + n^2 = 1$.

3. Vector Addition and Multiplication by Scalars

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

$$\vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}$$

$$k\vec{a} = ka_1\hat{i} + ka_2\hat{j} + ka_3\hat{k}$$

Properties of Addition

• Commutative: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
• Associative: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
• Triangle law: $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$

Section Formula

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.

4. Scalar (Dot) Product

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:

$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$$

Key Facts

• $\vec{a} \cdot \vec{a} = |\vec{a}|^2$
• $\vec{a} \cdot \vec{b} = 0$ if and only if the vectors are perpendicular (or one is zero).
• $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$.
• $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$, and $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$.
• The projection of $\vec{a}$ on $\vec{b}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$.

5. Vector (Cross) Product

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:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}$$

Key Facts

• $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$; the cross product is anti-commutative.
• $\vec{a} \times \vec{a} = \vec{0}$ for any vector.
• $\vec{a} \times \vec{b} = \vec{0}$ if and only if the vectors are parallel (or one is zero).
• $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$ (cyclic order).
• The area of the triangle formed by $\vec{a}$ and $\vec{b}$ is $\frac{1}{2}|\vec{a} \times \vec{b}|$.
• The cross product is distributive over addition but not commutative.

6. Scalar Triple Product

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.

Quick Revision Tables

Table 1: Vectors at a Glance

Table 2: Properties of Products

Table 3: Position and Section Formula

Mind Map

Important Diagrams (SVG)

Diagram 1: Triangle Law and Parallelogram Law of Addition

Diagram 2: Dot Product Angle and Cross Product Direction

Common Mistakes

1. Confusing the dot product (a scalar) with the cross product (a vector); the dot product can never be written with direction.
2. Forgetting that a x b = -(b x a); the cross product is anti-commutative, and switching the order flips the sign.
3. Writing a x a = |a|^2; in fact a x a = 0, the zero vector.
4. Using the cosine formula for the cross product magnitude; the cross product uses sin theta, the dot product uses cos theta.
5. Forgetting that the scalar triple product is zero exactly when the three vectors are coplanar.
6. In the section formula, mixing up m and n, or forgetting to divide by (m + n).
7. Writing the projection of a on b as (a.b)/|a|; it is (a.b)/|b|, dividing by the magnitude of the vector onto which projection is taken.
8. Confusing the direction cosines with the components; direction cosines are unitless, satisfy l^2 + m^2 + n^2 = 1, and equal the components divided by the magnitude.
9. Forgetting that for the cross product, the result is perpendicular to both vectors; checking this catches many errors.
10. Using the right-hand rule incorrectly; a x b curls from a toward b, and reversing the order reverses the direction of the unit normal.

Exam Tips

1. To test if two vectors are perpendicular, compute the dot product and set it to zero; to test parallelism, check that the cross product is the zero vector.
2. For the angle between two vectors, always use cos theta = (a.b)/(|a||b|) and verify theta lies in [0, pi].
3. Compute cross products using the determinant form with i, j, k in the first row; this avoids sign errors.
4. For area problems, remember that the area of a triangle is half the magnitude of the cross product of two of its sides.
5. When two vectors are given and a third unit vector perpendicular to both is required, compute (a x b)/|a x b|.
6. Verify the section formula result by testing with the midpoint (ratio 1:1), which must give (a + b)/2.
7. In component problems, compare corresponding components when equating vectors; a single differing component breaks equality.
8. Use the scalar triple product to test coplanarity of three vectors or four points by converting the points into three vectors.

Conclusion

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.