Class 12 Maths - Vectors: Concepts and Operations

Study Notes

Class 12 Maths - Vectors

In Class 12 Maths, one of the key areas of study is understanding the concept of "Vectors". This mathematical tool allows us to represent physical quantities such as displacement, velocity, and force by arrows or directed line segments in a two-dimensional plane. In this section, we'll explore what vectors are, how they are represented mathematically, their properties, and some common operations performed on them.

What Are Vectors?

A vector is a quantity characterized by both magnitude and direction. It can be thought of like an arrow in the plane which has both a length (magnitude) and points in a specific direction (direction). We often denote vectors with bold letters, like (\mathbf{a}), (\mathbf{b}), etc.

Representing Vectors Mathematically

Vectors can be represented using ordered pairs (x, y) in a Cartesian co-ordinate system:

[ \begin{align*} &\text{If } A(2,3) \ &\rightarrow \text{The vector }\mathbf{a} = 2\mathbf{i}-3\mathbf{j}, \ &\text{where } \mathbf{i}= \begin{pmatrix} 1\ 0 \end{pmatrix},\quad \mathbf{j}= \begin{pmatrix} 0\ 1 \end{pmatrix}. \end{align*} ]

Here, ( \mathbf{i} ) represents the horizontal component, and ( \mathbf{j} ) represents the vertical component of the vector. Another representation is through parametric equations:

[ \begin{align*} &\mathbf{r}(t)= \begin{pmatrix} x_0 + t\cos{\theta}\y_0+t\sin{\theta} \end{pmatrix} \end{align*} ]

This form shows the position of a point moving along a straight line from any point ( P(x_0,y_0) ) toward another point ( Q(x_0+d\cos{\theta},y_0+d\sin{\theta}) ).

Properties of Vectors

Some fundamental properties of vectors worth noting:

Addition: Vectors can be added by adding corresponding components. For example, if ( \mathbf{a}=\begin{pmatrix} 2\ 1 \end{pmatrix},\mathbf{b}=\begin{pmatrix} 3\ 4 \end{pmatrix} ), then ( \mathbf{a}+\mathbf{b}=\begin{pmatrix} 5\ 5 \end{pmatrix} ).
Scalar Multiplication: Scalar multiplication involves multiplying each component of a vector by a real number. If ( k ) is a scalar and ( \mathbf{a} ) is a vector, then ( k\mathbf{a}=\mathbf{a}+ \cdots+\mathbf{a} (k \text{ times}) ).
Magnitude and Direction: The magnitude of a vector (| \mathbf{v} |) is given by the Pythagorean Theorem: [ | \mathbf{v} |=\sqrt{(x^2+y^2)} ].
Unit Vector: A unit vector is a vector whose magnitude is equal to 1. For instance, a unit vector pointing north would have components (0, 1).

Common Operations Performed on Vectors

Here are three commonly performed operations on vectors:

Orthogonal Projection

Orthogonal projection is finding the shortest distance between a point and a vector. Consider a vector, say ( \mathbf{u} ) drawn from the origin O to point P on the terminal side of the vector AB. Then, the orthogonal projection of AB onto (\mathbf{u}) is AP.

Angle Between Two Vectors

The angle between two nonzero, nonparallel vectors is defined as the angle between their directions. Let ( \hat{\mathbf{a}} ) and ( \hat{\mathbf{b}} ) be unit vectors parallel to ( \mathbf{a} ) and ( \mathbf{b} ), respectively. Then, the cosine of the angle between ( \mathbf{a} ) and ( \mathbf{b} ) is given by: ( \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} ).

Cross Product

The cross product of two vectors gives a new vector perpendicular to both original ones. If ( \mathbf{a} ) and ( \mathbf{b} ) are two vectors, then their cross product is denoted by ( \mathbf{c}=\mathbf{a}\times \mathbf{b} ). The magnitude of ( \mathbf{c} ) can be found using the formula: ( | \mathbf{c}|=| \mathbf{a}| |\mathbf{b}| \sin(\theta) ), where ( \theta ) is the angle between ( \mathbf{a} ) and ( \mathbf{b} ).

Understanding these concepts will equip you well when dealing with problems involving vectors in your Class 12 Maths course.

Description

Explore the fundamental concepts of vectors in Class 12 Maths, including representation, properties like addition and scalar multiplication, common operations like orthogonal projection and cross product, and understanding the angle between two vectors. Enhance your knowledge of vectors for a strong foundation in mathematical understanding.