♦ Some Basic Concepts

♦ Types of Vectors

♦ Types of Vectors

`=>` Let ‘l’ be any straight line in plane or three dimensional space. This line can be given two directions by means of arrowheads. A line with one of these directions prescribed is called a directed line (Fig (i), (ii)).

`=>` Now observe that if we restrict the line l to the line segment AB, then a magnitude is prescribed on the line l with one of the two directions, so that we obtain a directed line segment (Fig (iii)). Thus, a directed line segment has magnitude as well as direction.

`color {blue} "Definition 1 : "` A quantity that has magnitude as well as direction is called a vector. Notice that a directed line segment is a vector (Fig (iii)), denoted as `bar(AB)` or simply as `bara` , and read as `‘"vector" bar(AB)` ’ or `‘"vector" bara ’`.

● The point A from where the vector `bar (AB)` starts is called its initial point, and the point B where it ends is called its terminal point.

●The distance between initial and terminal points of a vector is called the magnitude (or length) of the vector, denoted as `|bar(AB)|`, or `|bara |` , or a . The arrow indicates the direction of the vector.

`color{red} " Note "` Since the length is never negative, the notation `|bar a | < 0` has no meaning.

`color{red} "Position Vector"`

Consider a point `P` in space, having coordinates `(x, y, z)` with respect to the origin `O(0, 0, 0).` Then, the vector `bar (OP)` having O and P as its initial and terminal points, respectively, is called the position vector of the point `P` with respect to `O.`

`=>` Using distance formula, the magnitude of `bar (OP)` ( or ` bar r` ) is given by

`| vec(OP)| = sqrt ( x^2 +y^2 + z^2)`

`=>` The position vectors of points A, B, C, etc., with respect to the origin O are denoted by `vec a , vec b , vec c` etc.., respectively (Fig).

`color{red} "Direction Cosines"`

`=>` Consider the position vector `vec(OP)` (or `vec r` ) of a point P (x, y, z ) as in Fig .

`=>` The angles `α, β, γ` made by the vector `vecr` with the positive directions of x, y and z-axes respectively, are called its direction angles.

`=>` The cosine values of these angles, i.e., cosα, cosβ and cos γ are called direction cosines of the vector `vec r` , and usually denoted by l, m and n, respectively.

`=>` From Fig , one may note that the triangle OAP is right angled, and in it, we have `cos alpha = x /r` ( r stands for `|vecr |` )

`=>` Similarly, from the right angled triangles OBP and OCP, we may write `cos beta = y/r` and ` cos gamma = z /r` .

`=>` Thus, the coordinates of the point P may also be expressed as `(lr, mr,nr).` The numbers `lr, mr` and `nr`, proportional to the direction cosines are called as direction ratios of vector `vec r` , and denoted as a, b and c, respectively.

`color{red] "Key Concept"` One may note that `l^2 + m^2 + n^2 = 1` but `a^2 + b^2 + c^2 ≠ 1`, in general.

`=>` Now observe that if we restrict the line l to the line segment AB, then a magnitude is prescribed on the line l with one of the two directions, so that we obtain a directed line segment (Fig (iii)). Thus, a directed line segment has magnitude as well as direction.

`color {blue} "Definition 1 : "` A quantity that has magnitude as well as direction is called a vector. Notice that a directed line segment is a vector (Fig (iii)), denoted as `bar(AB)` or simply as `bara` , and read as `‘"vector" bar(AB)` ’ or `‘"vector" bara ’`.

● The point A from where the vector `bar (AB)` starts is called its initial point, and the point B where it ends is called its terminal point.

●The distance between initial and terminal points of a vector is called the magnitude (or length) of the vector, denoted as `|bar(AB)|`, or `|bara |` , or a . The arrow indicates the direction of the vector.

`color{red} " Note "` Since the length is never negative, the notation `|bar a | < 0` has no meaning.

`color{red} "Position Vector"`

Consider a point `P` in space, having coordinates `(x, y, z)` with respect to the origin `O(0, 0, 0).` Then, the vector `bar (OP)` having O and P as its initial and terminal points, respectively, is called the position vector of the point `P` with respect to `O.`

`=>` Using distance formula, the magnitude of `bar (OP)` ( or ` bar r` ) is given by

`| vec(OP)| = sqrt ( x^2 +y^2 + z^2)`

`=>` The position vectors of points A, B, C, etc., with respect to the origin O are denoted by `vec a , vec b , vec c` etc.., respectively (Fig).

`color{red} "Direction Cosines"`

`=>` Consider the position vector `vec(OP)` (or `vec r` ) of a point P (x, y, z ) as in Fig .

`=>` The angles `α, β, γ` made by the vector `vecr` with the positive directions of x, y and z-axes respectively, are called its direction angles.

`=>` The cosine values of these angles, i.e., cosα, cosβ and cos γ are called direction cosines of the vector `vec r` , and usually denoted by l, m and n, respectively.

`=>` From Fig , one may note that the triangle OAP is right angled, and in it, we have `cos alpha = x /r` ( r stands for `|vecr |` )

`=>` Similarly, from the right angled triangles OBP and OCP, we may write `cos beta = y/r` and ` cos gamma = z /r` .

`=>` Thus, the coordinates of the point P may also be expressed as `(lr, mr,nr).` The numbers `lr, mr` and `nr`, proportional to the direction cosines are called as direction ratios of vector `vec r` , and denoted as a, b and c, respectively.

`color{red] "Key Concept"` One may note that `l^2 + m^2 + n^2 = 1` but `a^2 + b^2 + c^2 ≠ 1`, in general.

`color{blue}{"Zero Vector :"}`

`=>` Zero Vector A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as `vec 0`.

`=>` Zero vector can not be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors `vec ("AA") , vec (BB)` represent the zero vector.

`color{blue}{"Unit Vector :"}`

`=>` Unit Vector A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector in the direction of a given vector `veca` is denoted by `hat a` .

`color{blue}{"Cointial Vector :"}`Coinitial Vectors Two or more vectors having the same initial point are called coinitial vectors.

`color{blue}{"Collinear Vector :"}` Collinear Vectors Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

`color{blue}{"Equal Vector :"}` Equal Vectors Two vectors `veca` and `vecb` are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as `vec a = vec b`

`color{blue}{"Negative Of Vector :"}`Negative of a Vector A vector whose magnitude is the same as that of a given vector (say, `vec(AB)` ), but direction is opposite to that of it, is called negative of the given vector.

For example, vector `vec (BA)` is negative of the vector `vec (AB)` , and written as `vec (BA) = - vec(AB)`

`color{blue} "Remark"` The vectors defined above are such that any of them may be subject to its parallel displacement without changing its magnitude and direction. Such vectors are called free vectors. Throughout this chapter, we will be dealing with free vectors only.

`=>` Zero Vector A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as `vec 0`.

`=>` Zero vector can not be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors `vec ("AA") , vec (BB)` represent the zero vector.

`color{blue}{"Unit Vector :"}`

`=>` Unit Vector A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector in the direction of a given vector `veca` is denoted by `hat a` .

`color{blue}{"Cointial Vector :"}`Coinitial Vectors Two or more vectors having the same initial point are called coinitial vectors.

`color{blue}{"Collinear Vector :"}` Collinear Vectors Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

`color{blue}{"Equal Vector :"}` Equal Vectors Two vectors `veca` and `vecb` are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as `vec a = vec b`

`color{blue}{"Negative Of Vector :"}`Negative of a Vector A vector whose magnitude is the same as that of a given vector (say, `vec(AB)` ), but direction is opposite to that of it, is called negative of the given vector.

For example, vector `vec (BA)` is negative of the vector `vec (AB)` , and written as `vec (BA) = - vec(AB)`

`color{blue} "Remark"` The vectors defined above are such that any of them may be subject to its parallel displacement without changing its magnitude and direction. Such vectors are called free vectors. Throughout this chapter, we will be dealing with free vectors only.

Q 3168767605

Represent graphically a displacement of `40` km, `30°` west of south.

Class 12 Chapter 10 Example 1

Class 12 Chapter 10 Example 1

The vector `vec(OP)` represents the required displacement

Q 3178767606

Classify the following measures as scalars and vectors.

(i) `5` seconds

(ii) `1000 cm^3`

(iii) `10` Newton (iv) `30 km//hr` (v) `10 g//cm^3`

(vi) `20 m//s` towards north

Class 12 Chapter 10 Example 2

(i) `5` seconds

(ii) `1000 cm^3`

(iii) `10` Newton (iv) `30 km//hr` (v) `10 g//cm^3`

(vi) `20 m//s` towards north

Class 12 Chapter 10 Example 2

(i) Time-scalar (ii) Volume-scalar (iii) Force-vector

(iv) Speed-scalar (v) Density-scalar (vi) Velocity-vector

Q 3188767607

In Fig 10.5, which of the vectors are:

(i) Collinear (ii) Equal (iii) Coinitial

Class 12 Chapter 10 Example 3

(i) Collinear (ii) Equal (iii) Coinitial

Class 12 Chapter 10 Example 3

(i) Collinear vectors ` : vec a , vec c ` and `vec d`

(ii) Equal vectors `: vec a` and `vec c`.

(iii) Coinitial vectors : `vec b, vec c` and `vec d`.