Vector Algebra Concepts for Class 12 Maths – A Quick Guide

Physical quantities are divided into two categories – Scalars and Vectors, Quantities that have only magnitude and no fixed direction are scalars e.g. speed, distance, mass, volume, temperature etc. and the ones having both magnitude and direction are called the vector quantities e.g. velocity, displacement, weight, force, momentum etc. Both types are essential in mathematics and physics for describing various quantities. This guide covers key vector algebra concepts, notations, and formulas for Class 12 Maths.

I strongly suggest downloading the hand-written pdf for the quick notes.

Table of Contents show

1. Basics of Vectors

A vector has an initial point and a terminal point. The vector from $A$ to $B$ is denoted as:
$\overrightarrow{AB}$
The magnitude (length) of a vector $\overrightarrow{a} = x \hat{i} + y \hat{j} + z \hat{k}$ is given by:
$| \overrightarrow{a} | = \sqrt{x^2 + y^2 + z^2}$

2. Position Vector, Direction Cosines & Ratios

Position vector of a point $P(x, y, z)$ is given by:
$\overrightarrow{OP} = x \hat{i} + y \hat{j} + z \hat{k}$

Direction cosines of any vector ( $l, m, n$ ) always satisfy the following relation:
$l^2 + m^2 + n^2 = 1$

The values of direction cosines i.e. ( $l, m, n)$ usually involve fractions or radicals. So we generally use the direction ratios (DRs), ( $a, b, c)$ , which are proportional to the direction cosines (DCs) while doing most of the problems of vector algebra. And these ratios simplify calculations in vector algebra.

3. Types of Vectors

There are many types of vectors which help describe different quantities. Some of them are:

Zero vector: $\overrightarrow{0}$

Unit vector: $\hat{a} = \frac{\overrightarrow{a}}{| \overrightarrow{a} |}$

Various types of vectors — *Various Types of Vectors*

Collinear vectors: $\overrightarrow{b} = \lambda \overrightarrow{a}$

Equal vectors: The ones having equal magnitude and same direction.

Download the pdf to learn more about these and some other types of Vectors.

4. Vector Operations

Just like in numbers, we can perform the operations like addition, subtraction etc. in vectors.

Addition: $\overrightarrow{a} + \overrightarrow{b}$ (Using Triangle/Parallelogram Law of Addition)

Subtraction: $\overrightarrow{a} - \overrightarrow{b} = \overrightarrow{a} + (-\overrightarrow{b})$

Scalar multiplication: $k \overrightarrow{a}$ scales the vector as per the value of the scalar k. Multiplying a vector by a negative scalar will change its direction also.

Multiplication in vector alegbra is defined in two ways, one is a scalar product and the other is a vector product as defined below. But before that, let us know about the component form as well.

5. Component Form of a Vector

A vector in 3D space:
$\overrightarrow{a} = x \hat{i} + y \hat{j} + z \hat{k}$

Magnitude:
$| \overrightarrow{a} | = \sqrt{x^2 + y^2 + z^2}$

Equal Vectors Condition: To check if two vectors are equal, their corresponding components must be identical i.e.

$x_1 = x_2, \quad y_1 = y_2, \quad z_1 = z_2$

Collinear Vectors Condition: Two vectors are collinear if and only if

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

6. Section Formulae

For a point dividing $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ in the ratio $m:n$ :
$(x, y, z) = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n}\right)$

Moreover, the Mid-point Formula can be deduced by taking the ratio as 1:1. The formulas can also be easily expressed in the form of vectors as you can see in the attached pdf. You can also watch the videos on various PYQs/MCQs here.

7. Dot Product (Scalar Product)

The dot product, also known as the scalar product of two vectors, is a very important tool to define many operations in vector algebra. It is defined by the following formula:

$\overrightarrow{a} \cdot \overrightarrow{b} = | \overrightarrow{a} | | \overrightarrow{b} | \cos \theta$

Here $\theta$ being the angle between the two vectors. And using this, we can easily get the condition for perpendicular and parallel or collinear vectors.

Perpendicular Vectors (Condition of Perpendicularity):
$\overrightarrow{a} \cdot \overrightarrow{b} = 0$

Collinear Vectors: In this case, we have, $\cos \theta = \pm1$ as per the value of angle $\theta$

8. Projection of a Vector

Scalar Projection: Scalar Projection of one vector on another is defined as the dot product of the vector and the unit vector along the vector on which we have to find the projection i.e.
$\text{Projection of } \overrightarrow{a} on \overrightarrow{b} = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{| \overrightarrow{b} |}$

Vector Projection: To find this, we have to just multiply the scalar projection (formula mentioned above) with the unit vector along the one on which we need to find the vector projection as following:
$\text{Projection vector of } \overrightarrow{a} on \overrightarrow{b} = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{| \overrightarrow{b} |^2} \overrightarrow{b}$

9. Cross Product (Vector Product) and its Properties

Just like the dot product, the cross product is another very important way to multiply two vectors in vector algebra and can be easily explained and calculated using the following formula:

Formula:
$\overrightarrow{a} \times \overrightarrow{b} = | \overrightarrow{a} | | \overrightarrow{b} | \sin \theta \hat{n}$

And as the name suggests, it is a vector quantity and its direction is given by the right-handed system. Further details are available in the attached PDF.

You can easily calculate the value of cross product in the component form using determinant as follows:

Determinant Form:
$\overrightarrow{a} \times \overrightarrow{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix}$

For Parallel Vectors:
$\overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{0}$

10. Miscellaneous Vector Algebra Concepts

Unit vector perpendicular to $\overrightarrow{a}$ and $\overrightarrow{b}$ :
$\hat{n} = \frac{\overrightarrow{a} \times \overrightarrow{b}}{|\overrightarrow{a} \times \overrightarrow{b}|}$

For Vectors equally inclined to three axes:
$(x, y, z)$ satisfy
$l = m = n = \pm \frac {1}{\sqrt{3}}$

Conclusion

This guide provides a concise summary of vector algebra for Class 12, covering fundamental concepts and formulas. Mastering these topics will surely help you score better in your board exams. It is highly recommended that you download this pdf and make your Maths Better not just by reading the notes, but also practicing the related questions.

Signing off for today, see you soon with another important topic i.e. 3D Geometry.

VECTOR ALGEBRA CONCEPTS Download

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Vector Algebra Concepts for Class 12 Maths – A Quick Guide

1. Basics of Vectors

2. Position Vector, Direction Cosines & Ratios

3. Types of Vectors

4. Vector Operations

5. Component Form of a Vector

6. Section Formulae

7. Dot Product (Scalar Product)

8. Projection of a Vector

9. Cross Product (Vector Product) and its Properties

10. Miscellaneous Vector Algebra Concepts

Conclusion

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1. Basics of Vectors

2. Position Vector, Direction Cosines & Ratios

3. Types of Vectors

4. Vector Operations

5. Component Form of a Vector

6. Section Formulae

7. Dot Product (Scalar Product)

8. Projection of a Vector

9. Cross Product (Vector Product) and its Properties

10. Miscellaneous Vector Algebra Concepts

Conclusion

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