Vector Basics

Vector Basics: Introduction to Vectors

Welcome to the world of vectors! In JEE Main, Vector Algebra and 3D Geometry form a crucial part of the syllabus, often carrying significant weightage. Understanding vectors is not just about memorizing formulas; it's about visualizing and applying them to solve real-world problems. Let's dive into the basics!

What is a Vector?

A vector is a mathematical object that has both magnitude (length) and direction. Think of it as an arrow pointing from one point to another. The magnitude represents how long the arrow is, and the direction tells us where it's pointing.

Scalars vs. Vectors

It's essential to distinguish between scalars and vectors.

• Scalars: These are quantities that have only magnitude (size). Examples include temperature, speed, mass, and time. A scalar is fully described by a single number and a unit.
• Vectors: These quantities have both magnitude and direction. Examples include displacement, velocity, acceleration, and force.

Consider this: saying "the temperature is 25°C" describes a scalar quantity, but saying "the car is moving at 60 km/h towards the north" describes a vector quantity because it includes both speed (magnitude) and direction.

Representation of Vectors

Vectors can be represented in two main ways:

1. Arrow Notation: A vector is represented by an arrow, where the length of the arrow corresponds to the magnitude of the vector and the arrowhead indicates the direction. We denote a vector as $\vec{a}$.
2. Component Form: In a two-dimensional (2D) or three-dimensional (3D) space, we can express a vector as an ordered list of numbers, called components. For example, in 3D space, a vector $\vec{a}$ can be written as $\vec{a} = (a_1, a_2, a_3)$. This means that the vector has components $a_1$ along the x-axis, $a_2$ along the y-axis, and $a_3$ along the z-axis.

Position Vector

A position vector specifies the location of a point with respect to the origin (0, 0, 0) of a coordinate system. If a point P has coordinates (x, y, z), then its position vector $\vec{r}$ is given by:

Here, $\hat{i}$, $\hat{j}$, and $\hat{k}$ are unit vectors along the x, y, and z axes, respectively. These unit vectors each have a length of 1.

For example, if P is (2, -3, 4), then the position vector $\vec{r} = 2\hat{i} - 3\hat{j} + 4\hat{k}$. Imagine starting at the origin and moving 2 units along the x-axis, -3 units along the y-axis, and 4 units along the z-axis to reach point P.

Types of Vectors

• Zero Vector: A vector with zero magnitude and arbitrary direction. It is denoted as $\vec{0} = (0, 0, 0)$.
• Unit Vector: A vector with a magnitude of 1. It's used to specify direction. We often use the "hat" notation to denote a unit vector, like $\hat{a}$.
• Equal Vectors: Two vectors are equal if they have the same magnitude and direction. This means their corresponding components are equal. If $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, then $\vec{a} = \vec{b}$ if and only if $a_1 = b_1$, $a_2 = b_2$, and $a_3 = b_3$.
• Negative Vector: The negative of a vector $\vec{a}$ is a vector with the same magnitude but opposite direction, denoted as $-\vec{a}$. If $\vec{a} = (a_1, a_2, a_3)$, then $-\vec{a} = (-a_1, -a_2, -a_3)$.
• Collinear Vectors: Vectors that lie on the same line or parallel lines are called collinear vectors. They are scalar multiples of each other. If $\vec{a}$ and $\vec{b}$ are collinear, then $\vec{a} = \lambda \vec{b}$ for some scalar $\lambda$.

Magnitude of a Vector

The magnitude of a vector $\vec{a} = (a_1, a_2, a_3)$ is its length, denoted as $|\vec{a}|$. It's calculated using the following formula (which is essentially an extension of the Pythagorean theorem):

The magnitude is always a non-negative scalar. For example, if $\vec{a} = (2, -3, 4)$, then $|\vec{a}| = \sqrt{2^2 + (-3)^2 + 4^2} = \sqrt{4 + 9 + 16} = \sqrt{29}$.

Unit Vector

To find the unit vector in the direction of a given vector $\vec{a}$, we divide the vector by its magnitude:

For example, if $\vec{a} = (2, -3, 4)$ and $|\vec{a}| = \sqrt{29}$, then the unit vector $\hat{a} = \frac{1}{\sqrt{29}}(2, -3, 4) = (\frac{2}{\sqrt{29}}, \frac{-3}{\sqrt{29}}, \frac{4}{\sqrt{29}})$. You can verify that the magnitude of $\hat{a}$ is indeed 1.

Direction Cosines and Direction Ratios

Direction cosines are the cosines of the angles that a vector makes with the positive directions of the coordinate axes. Let $\vec{a}$ make angles $\alpha$, $\beta$, and $\gamma$ with the x, y, and z axes, respectively. Then the direction cosines are:

A fundamental property of direction cosines is:

Derivation: Consider a vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. Then $l = \frac{x}{|\vec{r}|}$, $m = \frac{y}{|\vec{r}|}$, and $n = \frac{z}{|\vec{r}|}$. Squaring and adding these gives: $l^2 + m^2 + n^2 = \frac{x^2 + y^2 + z^2}{|\vec{r}|^2} = \frac{|\vec{r}|^2}{|\vec{r}|^2} = 1$.

Direction ratios are numbers proportional to the direction cosines. If $a, b, c$ are direction ratios, then:

This means that the direction cosines can be obtained by dividing the direction ratios by the magnitude of the vector formed by the direction ratios.