{"id":18470,"date":"2022-07-28T14:35:21","date_gmt":"2022-07-28T09:05:21","guid":{"rendered":"https:\/\/apboardsolutions.com\/?p=18470"},"modified":"2023-03-30T15:04:37","modified_gmt":"2023-03-30T09:34:37","slug":"inter-1st-year-maths-1a-addition-of-vectors-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.com\/inter-1st-year-maths-1a-addition-of-vectors-formulas\/","title":{"rendered":"Inter 1st Year Maths 1A Addition of Vectors Formulas"},"content":{"rendered":"

Use these Inter 1st Year Maths 1A Formulas<\/a> PDF Chapter 4 Addition of Vectors to solve questions creatively.<\/p>\n

Intermediate 1st Year Maths 1A Addition of Vectors Formulas<\/h2>\n

Scalar :<\/strong>
\nA physical quantity which has only magnitude is called a scalar quantity. All the real numbers will be taken as scalars.<\/p>\n

Vector :<\/strong>
\nA physical quantity which has both magnitude and direction.
\ne.g. : Velocity acceleration, force, momentum.<\/p>\n

Position vector :<\/strong>
\nLet ‘O’ and ‘P be any points in space. Then OP is called the position vector of the point ‘P w.r.t. origin ‘O’.
\nNote : \\(\\overline{A B}\\) = Position vector of B- position vector of A’
\n= \\(\\overline{O B}-\\overline{O A}\\)<\/p>\n

Coinitial vector:<\/strong>
\nVectors having the same initial point are called coinitial vectors.
\ne.g. : \\(\\overline{O A}, \\overline{O B}, \\overline{O C}\\) etc.<\/p>\n

Unit vector:<\/strong>
\nA vector whose magnitude is one-unit is called unit vector.
\nUnit vector in the direction of a is denoted by a\u0302 = \\(\\frac{\\bar{a}}{|a|}\\).
\nFor any non-zero vector \\(\\bar{a}=|\\bar{a}|\\) a\u0302.<\/p>\n

\"Inter<\/p>\n

Like vectors :<\/strong>
\nIf two vectors are parallel and having the same direction then they are called like vectors.<\/p>\n

Unlike vectors :<\/strong>
\nIf two vectors are parallel and having Opposite direction then they are called unlike vectors.
\nThe position vector of any point C on \\(\\overline{A B}\\) can be taken as \u03bb\\(\\bar{a}\\) + \u00b5\\(\\bar{a}\\), where \u03bb + \u00b5 = 1<\/p>\n

Angle between vectors:<\/strong>
\nLet \\(\\overline{O A}=\\bar{a}\\) = a, \\(\\overline{O B}=\\bar{a}\\) = b be any two non – zero vectors, then angle AOB is defined as angle between vectors \\(\\bar{a}, \\bar{b}\\) and is denoted by \\((\\bar{a}, \\bar{b})\\) where 0 \u2264 \\((\\bar{a}, \\bar{b})\\) \u2264 180\u00b0.<\/p>\n

Addition of vectors (or) Parallelogram Law:<\/strong>
\nIf \\(\\bar{a}, \\bar{b}\\) are the adjacent sides of a parallelogram the diagonals which is coinitial with \\(\\bar{a}, \\bar{b}\\) is given by \\(\\bar{a}+ \\bar{b}\\) and its magnitude is given by
\n\\(|\\bar{a}+\\bar{b}|=\\sqrt{|\\bar{a}|^{2}+|\\bar{b}|^{2}+2|\\bar{a}||\\bar{b}| \\cos (\\bar{a}, \\bar{b})}\\)<\/p>\n

Triangle law:<\/strong>
\nIf two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their sum is represented by the third side taken in the reverse order.<\/p>\n

Note : Addition of vectors is of 2 types :<\/p>\n

    \n
  1. Commutative<\/li>\n
  2. Associative.<\/li>\n<\/ol>\n

    (ie) (i) \\(\\bar{a}+\\bar{b}=\\bar{b}+\\bar{a}\\) and (ii) \\((\\bar{a}+\\bar{b})+\\bar{c}=\\bar{a}+(\\bar{b}+\\bar{c})\\)
    \nRule: \\(|\\bar{a}| \\sim|\\bar{b}| \\leq|\\bar{a}-\\bar{b}| \\leq|\\bar{a}+\\bar{b}| \\leq|\\bar{a}|+|\\bar{b}|\\)<\/p>\n

    Parallel (or) Collinear vector:<\/strong><\/p>\n

      \n
    1. Two vectors a, b are parallel or collinear, then a = \u03bb. b, where ‘\u03bb’ is a scalar.<\/li>\n
    2. If \\(\\bar{a}\\) = (a1<\/sub>, a2<\/sub>, a3<\/sub>), b = (b1<\/sub>, b2<\/sub>, b3<\/sub>) are parallel or collinear, then
      \n\\(\\frac{a_{1}}{b_{1}}=\\frac{a_{2}}{b_{2}}=\\frac{a_{3}}{b_{3}}\\)
      \nNote : Zero vector is parallel to any vector.<\/li>\n
    3. If three points with position vectors a, b, c are to be collinear. The necessary and sufficient condition is that there exists scalars \\(\\bar{a}, \\bar{b}, \\bar{c}\\) not all zero, such that
      \nl\\(\\bar{a}\\) + m\\(\\bar{b}\\) + n\\(\\bar{c}\\) = 0, l + m + n = 0.<\/li>\n
    4. If \\(\\bar{a}, \\bar{b}, \\bar{c}\\) are non-zero, non-collinear vectors such that l\\(\\bar{a}\\) + m\\(\\bar{b}\\) + n\\(\\bar{c}\\) = 0, then l = 0, m = 0, n = 0.<\/li>\n<\/ol>\n

      Linear combination of vectors :<\/strong>
      \nA linear combination of the system of vectors \\(\\bar{a}_{1}, \\bar{a}_{2}, \\ldots, \\bar{a}_{n}\\) is a vector.
      \nr = x1<\/sub>\\(\\bar{a}_{1}\\) + x2<\/sub>\\(\\bar{a}_{2}\\) + x3<\/sub>\\(\\bar{a}_{3}\\) + ………………. + xn<\/sub>\\(\\bar{a}_{n}\\)
      \nwhere x1<\/sub> x2<\/sub>, x3<\/sub>, ………………., xn<\/sub> are scalars.<\/p>\n

      Coplanar vectors:<\/strong>
      \nIf three or more vectors lie in the same plane (or) parallel to the same plane then they are called coplanar vectors. If one vector can be expressed as a linear combination of the remaining vectors, then the vectors are coplanar vectors.
      \nIf \\(\\bar{a}\\) = x\\(\\bar{b}\\) + y\\(\\bar{c}\\), where x, y, are scalars, then \\(\\bar{a}, \\bar{b}, \\bar{c}\\) are coplanar.<\/p>\n

      \"Inter<\/p>\n

      Linearly dependent system of vectors :<\/strong>
      \nA system of vectors \\(\\bar{a}_{1}, \\bar{a}_{2}, \\bar{a}_{3}, \\ldots \\ldots, \\bar{a}_{n}\\) is said to
      \nbe linearly dependent, if there exists a system of scalars x1<\/sub>, x2<\/sub>, x3<\/sub>, ……………….., xn<\/sub> not all zero
      \nsuch that
      \nx1<\/sub>\\(\\bar{a}_{1}\\) + x2<\/sub>\\(\\bar{a}_{2}\\) + x3<\/sub>\\(\\bar{a}_{3}\\) + ………………. + xn<\/sub>\\(\\bar{a}_{n}\\) = \\(\\bar{0}\\)<\/p>\n