UNIT-IV

CHAPTER

VECTORS & THREE- DIMENSIONAL VECTORS

GEOMETRY

Syllabus

Vectors and Scalars, Magnitude and direction of a vector. Direction cosines and direction ratios of a vector, "types of vectors" equal, unit, zero, parallel and collinear vectors, position vector of a "point, negative of a vector", components of a vector, addition of vectors (properties of addition, laws of addition), Multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical interpretation, properties and application of Scalar (dot) product of vectors, Vector (cross) product of vectors, scalar triple product of vectors.

Chapter Analysis

2017 2016 2018

TOPIC Delhi Delhi OD OD Delhi/OD

Properties 2 Q. (1 Mark) 2 Q.

Angle between vectors 1 Q. (1 Mark)

(4 Marks) 2 Q.

(2 Marks)

Dot Product _ _ _

Cross product 2 Q. 1 Q.

(4 Marks)

1 Q.

Area of triangle (4 Marks) (4 Marks)

Coplanarity 1 Q. 1 Q. 1 Q.

(4 Marks) (4 Marks) (4 Marks)

TOPIC-1

Basic Algebra of Vectors TOPIC - 1 TOPIC - 2 Dot Product of Vectors Page 366 Page 378

Basic Algebra of Vectors

TOPIC - 3 Page 389

Cross Product

Revision Notes

TOPIC - 4 Page 403

Scalar Triple Product

1. Vector: Basic Introduction:

A quantity having magnitude as well as the direction is called a vector. It is denoted as <math>\vec{AB}</math> or <math>\vec{a}</math>. Its magnitude (or modulus) is <math>|\overrightarrow{AB}|</math> or <math>|\overrightarrow{a}|</math> otherwise, simply AB or a.

• Vectors are denoted by symbols such as <math>\vec{a}</math>. [Pictorial representation of vector]

1. Initial and Terminal Points:

The initial and terminal points means that point from which the vector originates and terminates respectively.

367 VECTORS

3. Position Vector:

The position vector of a point say <math>P(x, y, z)</math> is <math>\overrightarrow{OP} = \overrightarrow{r} = x\hat{i} + y\hat{j} + z\hat{k}</math> and the magnitude is <math>|\vec{r}| = \sqrt{x^2 + y^2 + z^2}</math>. The vector <math>\overrightarrow{OP} = \overrightarrow{r} = x\hat{i} + y\hat{j} + z\hat{k}</math> is said to be in its component form. Here x, y, z are called the scalar components or rectangular components of <math>\vec{r}</math> and <math>x\hat{i}</math>, <math>y\hat{j}</math>, <math>z\hat{k}</math> are the vector components of <math>\vec{r}</math> along x, y, z-axis respectively.

• Also, <math>\overrightarrow{AB} = (\text{Position Vector of } B) - (\text{Position Vector of } A)</math>. For example, let <math>A(x_1, y_1, z_1)</math> and <math>B(x_2, y_2, z_2)</math>. Then, <math>\vec{AB} = (x_2\hat{i} + y_2\hat{j} + z_2\hat{k}) - (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}).</math>

Here <math>\hat{i}</math>, <math>\hat{j}</math> and <math>\hat{k}</math> are the unit vectors along the axes OX, OY and OZ respectively (The discussion about unit vectors is given later under 'types of vectors').

4. Direction Ratios and Direction Cosines:

If <math>\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}</math>, then coefficient of <math>\hat{i}</math>, <math>\hat{j}</math>, <math>\hat{k}</math> in <math>\vec{r}</math> i.e., x, y, z are called the direction ratios (abbreviated as d.r.'s) of vector <math>\vec{r}</math>. These are denoted by a, b, c (i.e., <math>a = x</math>, <math>b = y</math>, <math>c = z</math>; in a manner we can say that scalar components of vector r and its d.r.'s both are the same).

Also, the coefficients of <math>\hat{i}</math>, <math>\hat{j}</math>, <math>\hat{k}</math> in <math>\overset{\rightarrow}{r}</math> (which is the unit vector of <math>\overset{\rightarrow}{r}</math>) i.e., <math>\frac{x}{\sqrt{x^2+y^2+z^2}}</math>, <math>\frac{y}{\sqrt{x^2+y^2+z^2}}</math>, <math>\frac{z}{\sqrt{x^2+y^2+z^2}}</math> are called direction cosines (which is abbreviated as d.c.'s) of vector

These direction cosines are denoted by l, m, n such that <math>l = \cos \alpha</math>, <math>m = \cos \beta</math>, <math>n = \cos \gamma</math> and <math>l^2 + m^2 + n^2 = 1</math> <math>\Rightarrow \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1.</math>

It can be easily concluded that <math>\frac{x}{r} = l = \cos \alpha</math>, <math>\frac{y}{r} = m = \cos \beta</math>, <math>\frac{z}{r} = n = \cos \gamma</math>.<br>

Therefore, <math>\vec{r} = lr\hat{i} + mr\hat{j} + nr\hat{k} = r(\cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k})</math>. [Here <math>r = |\vec{r}|</math>].

5. Addition of vectors

1. Triangular law: If two adjacent sides (say sides AB and BC) of a triangle ABC are represented by <math>\vec{a}</math> and <math>\vec{b}</math> taken in same order, then the third side of the triangle taken in the reverse order gives the sum of vectors <math>\vec{a}</math> and <math>\vec{b}</math> i.e., <math>\vec{AC} = \vec{AB} + \vec{BC} \Rightarrow \vec{AC} = \vec{a} + \vec{b}</math>.

Also since <math>\overrightarrow{AC} = -\overrightarrow{CA} \Rightarrow \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \overrightarrow{0}</math>.

And <math>\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{AC} = 0 \Rightarrow \overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = 0</math>

1. Parallelogram law: If two vectors <math>\vec{a}</math> and <math>\vec{b}</math> are represented in magnitude and the direction by the two adjacent sides (say AB and AD) of a parallelogram ABCD, then their sum is given by that diagonal of parallelogram which is co-initial with <math>\vec{a}</math> and <math>\vec{b}</math> i.e., <math>\vec{OC} = \vec{OA} + \vec{OB}</math>.

6. Properties of Vector Addition

1. Commutative property: <math>\vec{a} + \vec{b} = \vec{b} + \vec{a}</math>

Consider <math>\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}</math> and <math>\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}</math> be any two given vectors, then <math>\vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k} = \vec{b} + \vec{a}</math>.

1. Associative property: <math>(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})</math>.
2. Additive identity property : <math>\vec{a} + \vec{0} = \vec{0} + \vec{a} = \vec{a}</math>.
3. Additive inverse property : <math>\vec{a} + (-\vec{a}) = \vec{0} = (-\vec{a}) + \vec{a}</math>.

Note: Multiplication of a vector by a scalar

Let <math>\vec{a}</math> be any vector and <math>\vec{k}</math> be any scalar. Then the product <math>\vec{ka}</math> is defined as a vector whose magnitude is <math>|\vec{k}|</math> times that of <math>\vec{a}</math> and the direction is

1. same as that of a if k is positive, and

(ii) opposite as that of a if k is negative.