Understanding Vector Algebra: A Student’s Guide

Vector algebra is the branch of mathematics that studies vectors, their properties, and operations. It is essential in understanding physical quantities that possess both magnitude and direction. The subject is fundamental for various applications in geometry and physics, including calculations involving displacement, force, and velocity.

Scalars and Vectors: Definitions

A scalar is a quantity defined completely by its magnitude alone. Examples include speed, distance, and mass. In contrast, a vector is a quantity defined by both magnitude and direction, such as velocity or displacement.

If a body moves 8 km due north, its displacement is a vector because it includes the direction. If only the quantity 8 km is given, it represents a scalar value without direction. Understanding this distinction is critical in vector algebra.

Geometrical Representation of Vectors

A vector is represented as a directed line segment in space. The length denotes magnitude, and the arrow shows direction. The initial and terminal points of the segment indicate its start and end locations, respectively.

Position vectors express points in space relative to the origin. If $A(x_1, y_1, z_1)$, the position vector is $\vec{a} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}$. Every point has a unique position vector from the origin in three-dimensional space.

Types and Properties of Vectors

Vectors are classified based on magnitude and direction relations. Equal vectors share both magnitude and direction, while negative vectors have the same magnitude but opposite direction. The unit vector has a magnitude of one and indicates direction only.

• Equal vectors have identical direction
• Negative vectors are oppositely directed
• Unit vector’s magnitude is always one

The unit vector $\hat{a}$ corresponding to a vector $\vec{a}$ is given by $\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}$, where $|\vec{a}|$ denotes the magnitude. Unit vectors along coordinate axes are denoted by $\hat{i}$, $\hat{j}$, and $\hat{k}$.

Vector Operations and Algebraic Rules

Scalar multiplication, vector addition, and subtraction form the basis of vector algebra. Multiplying a vector by a scalar changes only its magnitude, not direction, unless the scalar is negative, which reverses direction.

Vectors add according to the triangle law: if $\vec{a}$ and $\vec{b}$ are vectors, their sum is the vector from the tail of $\vec{a}$ to the tip of $\vec{b}$ placed consecutively. This is foundational to the Triangle Law of Vector Addition.

• Addition is commutative: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
• Addition is associative: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
• Multiplication by scalar: $k \vec{a}$, $k \in \mathbb{R}$

Subtraction of vectors is defined as addition with the negative vector: $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$. The result is a vector directed from the tip of $\vec{b}$ to the tip of $\vec{a}$.

Direction cosines are the cosines of angles made by a vector with the positive x, y, and z axes. For a vector $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$, direction cosines are $\cos\alpha = \dfrac{a_1}{|\vec{a}|}$, $\cos\beta = \dfrac{a_2}{|\vec{a}|}$, $\cos\gamma = \dfrac{a_3}{|\vec{a}|}$.

Section Formula and Collinearity

The section formula gives the position vector of a point dividing a segment in a given ratio. If $P$ divides $AB$ in the ratio $m:n$, the position vector is $\vec{p} = \dfrac{m\vec{b} + n\vec{a}}{m+n}$, where $\vec{a}$, $\vec{b}$ are position vectors of $A$, $B$.

Collinearity of vectors indicates that vectors are parallel. For collinearity, vectors must satisfy $\vec{a} = k\vec{b}$ for some scalar $k$. The Addition of Vectors also assists in verifying collinearity.

Scalar (Dot) Product and Properties

The scalar (dot) product of two vectors is defined as $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. The result is a scalar. If vectors are perpendicular, their scalar product is zero.

• Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
• Distributive over addition: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
• Zero if vectors are orthogonal

Projection of a vector $\vec{a}$ onto $\vec{b}$ is $|\vec{a}| \cos\theta$, representing the component of $\vec{a}$ in the direction of $\vec{b}$. This is vital in physics for determining effective components of forces and velocities.

Vector (Cross) Product and Applications

The vector (cross) product is defined by $\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin\theta \;\hat{n}$, where $\hat{n}$ is a unit vector perpendicular to the plane of $\vec{a}$ and $\vec{b}$. The result is a vector. It is anti-commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$.

• Magnitude: $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}| \sin\theta$
• Direction given by right-hand rule
• Zero if vectors are parallel

The cross product is used in calculating area. For a parallelogram with adjacent sides $\vec{a}$, $\vec{b}$, the area is $|\vec{a} \times \vec{b}|$. For a triangle, the area is $\dfrac{1}{2} |\vec{a} \times \vec{b}|$. More on related products is in Vector Triple Product.

Vector Algebra Examples and Identities

Example: Given $\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - 4\hat{j} + 5\hat{k}$. The dot product is $\vec{a} \cdot \vec{b} = (2 \times 1) + (3 \times -4) + (1 \times 5) = 2 - 12 + 5 = -5$.

As an identity, for any vectors $\vec{a}$, $\vec{b}$ and scalar $k$, $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$. Common formulas and rules can be further reviewed with a Scalar Product of Vectors focus.