`=>` 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.
`=>` 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.