{"id":10243,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10243"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"vector-fields","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/vector-fields\/","title":{"rendered":"Vector Fields"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Circular Motion and Gravitation<\/strong><\/h2>\n

Chapter: Vector fields<\/strong><\/h3>\n

Reference: AP Physics Algebra, <\/em>Circular Motion and Gravitation, <\/em>Vector fields, <\/em>Scalar, Vector, <\/em>Representation of A Vector Quantity<\/em>, <\/em>Position Vector<\/em>, <\/em>Types of Vectors<\/em>, <\/em>Vector Addition<\/em>, <\/em>Triangle law of vector addition, <\/em>Vector Subtraction<\/em>, <\/em>Multiplication of a Vector by a Scalar<\/em>, <\/em>Scalar (or dot) product of two vectors<\/em>, <\/em>Two important properties of the scalar product<\/em><\/p>\n

After studying this chapter, you should be able to:<\/strong><\/p>\n

    \n
  • Know the concept of the vector<\/li>\n
  • Calculate vector addition, subtraction, multiplication<\/li>\n
  • Know the important properties of vector<\/li>\n<\/ul>\n

    Scalar<\/strong><\/p>\n

    The physical quantity which has a magnitude but no specific direction.<\/p>\n

    e.g. \u2013 distance, mass, speed.<\/p>\n

    Vector<\/strong><\/p>\n

    The physical quantity which has a magnitude as well as direction and follows the vector law of addition. e.g. \u2013 forces, velocity, displacement, momentum.<\/p>\n

    Note-1<\/strong><\/p>\n

    Current is not a vector quantity though it has direction and magnitude as it does not follow the vector law of addition.<\/p>\n

    Representation of A Vector Quantity<\/strong><\/p>\n

    A vector is represented by drawing an arrow proportional in length to the physical quantity being represented.<\/p>\n

      \n
    • A vector variable is represented by an arrow over the English or Greek alphabet.<\/li>\n
    • The same alphabet without the vector sign represents the magnitude of the vector and the same alphabet with a cap sign represents the direction of the vector.<\/li>\n<\/ul>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      \u00a0<\/p>\n

      \u00a0<\/p>\n

      \u00a0<\/p>\n

      Position Vector<\/strong><\/p>\n

      The position vector is used to specify the position of a certain body. The position vector of an object is measured from the origin, in general.<\/p>\n

      \u00a0<\/p>\n

      Types of Vectors:<\/strong><\/p>\n

      Zero Vector: <\/strong>A vector, whose initial and terminal points coincide, is called a zero vector or (null vector). It is denoted by 0.<\/p>\n

      A zero vector cannot be assigned a definite direction as it has zero magnitudes.<\/p>\n

      The vectors AA<\/em>,\u00a0 BB<\/em>\u00a0represent the zero vector.<\/p>\n

      Unit Vector: <\/strong>A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector.<\/p>\n

      The unit vector in the direction of a given vector a<\/em>\u00a0is denoted by a<\/em>.<\/p>\n

      Coinitial Vectors:<\/strong> Two or more vectors having the same initial point are called coinitial vectors.<\/p>\n

      Collinear Vectors:<\/strong> Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.<\/p>\n

      For example: Consider 3 vectors as shown in the figure, they all are parallel to each other but their magnitudes are different as well as the directions. But they are said to be collinear vectors because they are parallel to each other.<\/p>\n

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      Source: Kapdec.com<\/p>\n<\/div>\n

      Equal Vectors:<\/strong> Two vectors a<\/em>\u00a0and b<\/em>\u00a0are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and are written as a<\/em>\u00a0and b<\/em>.<\/p>\n

      For example: Consider 2 vectors whose magnitudes and their directions are the same irrespective of origin, then they are known as equal vectors.<\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Negative of a Vector:<\/strong> A vector whose magnitude is the same as that of a given vector but whose direction is opposite to that of it is called the negative of the given vector.<\/p>\n

      \u00a0<\/p>\n

      \"\"<\/p>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      \u00a0<\/p>\n

      Vector Addition<\/strong><\/p>\n

      A variety of mathematical operations can be performed with and upon\u00a0vectors<\/strong>. One such operation is the\u00a0addition<\/strong>\u00a0of\u00a0vectors<\/strong>. Two\u00a0vectors<\/strong>\u00a0can be added together to determine the result (or resultant). This process of adding two or more\u00a0vectors<\/strong>\u00a0has already been discussed in an earlier unit.<\/p>\n

      The two vectors a and b can be added giving the sum to be a + b. This requires joining them head to tail.<\/p>\n

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      Source: Kapdec.com<\/p>\n<\/div>\n

      Characteristics of Vector Math Addition<\/strong><\/p>\n

        \n
      • Commutative Law:<\/strong> a + b = b + a<\/li>\n
      • Associative law:<\/strong> (a + b) + c = a + (b + c)<\/li>\n<\/ul>\n

        \u00a0<\/p>\n

        \u00a0<\/p>\n

        Triangle law of vector addition<\/strong><\/p>\n

        The triangle law of vector addition<\/strong> is appropriate to deal with such a situation. If two vectors<\/strong>\u00a0are represented by two sides of a\u00a0triangle<\/strong> in sequence, then the third closing side of the triangle<\/strong>, in the opposite direction of the sequence, represents the\u00a0sum<\/strong>\u00a0(or resultant) of the two\u00a0vectors<\/strong>\u00a0in both magnitude and direction.<\/p>\n

        A vector AB<\/em>\u00a0simply means the displacement from point A to point B. Now consider a situation in that a girl moves from A to B and then from B to C.<\/p>\n

        \"\"<\/p>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        The net displacement made by the girl from point A to point C, is given by the vector AC<\/em>\u00a0and expressed as<\/p>\n

        \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 AC<\/em>=<\/em>AB<\/em>+BC<\/em><\/p>\n

        Vector Subtraction<\/strong><\/h3>\n

        To\u00a0subtract\u00a0two\u00a0vectors, you put their feet (or tails, the non-pointy parts) together; then draw the resultant\u00a0vector, which is the difference of the two\u00a0vectors, from the head of the\u00a0vector\u00a0you’re\u00a0subtracting\u00a0to the head of the\u00a0vector\u00a0you’re\u00a0subtracting\u00a0it from.<\/p>\n

        \u00a0<\/p>\n

        A reverse vector (-a) which is opposite of (a) has a similar magnitude as (a) but is pointed in the opposite direction.<\/p>\n

        \"\"<\/p>\n

        Source: Kapdec.com<\/p>\n<\/div>\n

        Multiplication of a Vector by a Scalar<\/strong><\/p>\n

        The multiplication of a vector by a scalar quantity is called “Scaling.” In this type of multiplication, only the magnitude of a vector is changed not the direction.<\/p>\n

          \n
        • S(a + b) = Sa + Sb<\/li>\n
        • (S + T)a = Sa + Ta<\/li>\n
        • a.1 = a<\/li>\n
        • a.0 = 0<\/li>\n
        • a.(-1) = -a<\/li>\n<\/ul>\n

          \u00a0<\/p>\n

          Vector joining two points<\/strong><\/p>\n

          If P1<\/sub>(x1<\/sub>, y1<\/sub>, z1<\/sub>) and P2<\/sub>(x2<\/sub>, y2<\/sub>, z2<\/sub>) are any two points, then the vector joining P1<\/sub> and P2<\/sub> is the vector P<\/em>1<\/em>P2<\/em>.<\/p>\n

          \"\"<\/p>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          Joining the points P1<\/sub> and P2<\/sub> with the origin O, and applying triangle law, from the triangle OP1<\/sub>P2<\/sub>, we have<\/p>\n

          OP<\/em>1<\/em>+<\/em>P<\/em>1<\/em>P<\/em>2<\/em>=<\/em>O<\/em>P2<\/em>\u00a0<\/p>\n

          Using the properties of vector addition, the above equation becomes<\/p>\n

          P<\/em>1<\/em>P<\/em>2<\/em>=<\/em>OP<\/em>2<\/em>–<\/em>O<\/em>P1<\/em>\u00a0<\/p>\n

          \"\"<\/p>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          The magnitude of vector P<\/em>1<\/em>P2<\/em>\u00a0is given by<\/p>\n

          \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 P<\/em>1<\/em>P<\/em>2<\/em>=<\/em>(<\/em>x<\/em>2<\/em>–<\/em>x<\/em>1<\/em>)<\/em>2<\/em>+<\/em>(<\/em>y<\/em>2<\/em>–<\/em>y<\/em>1<\/em>)<\/em>2<\/em>+<\/em>(<\/em>z<\/em>2<\/em>–<\/em>z<\/em>1<\/em>)2<\/em><\/p>\n

          Scalar (or dot) product of two vectors<\/strong><\/p>\n

          In mathematics, the\u00a0dot product<\/strong>\u00a0or\u00a0scalar product<\/strong>\u00a0is an algebraic operation that takes\u00a0two<\/strong>\u00a0equal-length sequences of numbers (usually coordinate\u00a0vectors<\/strong>), and returns a single number. … Geometrically, it is the\u00a0product<\/strong>\u00a0of the Euclidean magnitudes of the\u00a0two vectors<\/strong>\u00a0and the cosine of the angle between them.<\/p>\n

          The scalar product of two nonzero vectors a<\/em>\u00a0and b<\/em>, denoted by a<\/em> . b<\/em>, is defined as<\/p>\n

          \"\"<\/p>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          a<\/em> . <\/em>b<\/em>=<\/em>a<\/em>b cos\u03b8,<\/em><\/p>\n

          Two important properties of the scalar product<\/strong><\/p>\n

          Property 1<\/strong> (Distributivity of scalar product over addition) Let a<\/em>, b<\/em>\u00a0and c<\/em>\u00a0be any three vectors, then a<\/em> .<\/em>b<\/em>+<\/em>c<\/em>=<\/em>a<\/em> . <\/em>b<\/em>+<\/em>a<\/em> . c<\/em><\/p>\n

          Property 2<\/strong> Let a<\/em>\u00a0and b<\/em>\u00a0be any two vectors, and l<\/em> be any scalar. Then<\/p>\n

          \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 la<\/em>.<\/em>b<\/em>=<\/em>la<\/em>.<\/em>b<\/em>=<\/em>la<\/em> . <\/em>b<\/em>=<\/em>a<\/em> .<\/em>lb<\/em><\/p>\n

          Key points: <\/strong><\/p>\n

            \n
          • The position vector of a point P(x, y, z) is given as OP<\/em>=<\/em>r<\/em>=x<\/em>i<\/em>+y<\/em>j<\/em>+z<\/em>k,<\/em>\u00a0and its magnitude by x<\/em>2<\/em>+<\/em>y<\/em>2<\/em>+<\/em>z2<\/em>\u00a0.<\/li>\n
          • The scalar components of a vector are its direction ratios and represent its projections along the respective axes.<\/li>\n
          • The magnitude (r), direction ratios (a, b, c) and direction cosines<\/li>\n
          • (l<\/em>, m, n) of any vector are related as:<\/li>\n<\/ul>\n

            l=<\/em>a<\/em>r<\/em> ,\u00a0 m=<\/em>b<\/em>r<\/em> ,\u00a0 n=<\/em>cr<\/em><\/p>\n

              \n
            • The vector sum of the three sides of a triangle taken in order is 0<\/em>.<\/li>\n
            • The vector sum of two coinitial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.<\/li>\n
            • The multiplication of a given vector by a scalar l, changes the magnitude of the vector by the multiple |l|, and keeps the direction the same (or makes it opposite) according to the value of l is positive (or negative).<\/li>\n
            • For a given vector a<\/em>, the vector a<\/em>=<\/em>a<\/em>|<\/em>a|<\/em>\u00a0gives the unit vector in the direction of a<\/em>.<\/li>\n
            • The position vector of a point R dividing a line segment joining<\/li>\n
            • the points P and Q whose position vectors a<\/em>\u00a0and b<\/em>\u00a0are respectively, in the ratio m : n<\/li>\n<\/ul>\n

              (i) \u00a0\u00a0\u00a0 internally, is given by n<\/em>a<\/em>+m<\/em>bm+n<\/em>.<\/p>\n

              (ii) \u00a0\u00a0\u00a0 externally, is given by m<\/em>b<\/em>-n<\/em>am-n<\/em>.<\/p>\n

                \n
              • The scalar product of two given vectors a<\/em>\u00a0and b<\/em>\u00a0having angle q between them is defined as<\/li>\n<\/ul>\n

                \u00a0<\/p>\n

                Also, when a<\/em> . b<\/em>\u00a0is given, the angle \u2018q\u2019 between the vectors may be determined by<\/p>\n

                \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 cos\u03b8=<\/em>a<\/em> . <\/em>b<\/em>a<\/em>|<\/em>b|<\/em><\/p>\n

                \u00a0<\/p>\n

                  \n
                • If q is the angle between two vectors a<\/em>\u00a0and b<\/em>, then their cross product is given as<\/li>\n<\/ul>\n

                  a<\/em>\u00d7<\/em>b<\/em>=<\/em>a<\/em>b<\/em>sin\u03b8 <\/em>n<\/em><\/p>\n

                  where n<\/em>\u00a0is a unit vector perpendicular to the plane containing a<\/em>\u00a0and b<\/em>. Such that a<\/em>, <\/em>b<\/em>, n<\/em>\u00a0form a right-handed system of coordinate axes.\u00a0<\/p>\n

                  \u00a0<\/p>\n

                  <\/p>\n