Some Basic Concepts and Types of Vectors

  • A quantity that has magnitude as well as direction is called a Vector
  • The distance between initial (starting) and terminal (ending) points of a vector is called the magnitude (or length).
  • If \overrightarrow{r} = x\hat{i} + y\hat{j} + z\hat{k} is not a unit vectors then x, y, z are called scalar components or direction ratios of a vector \overrightarrow{r} and its direction cosines are \frac{x}{|\overrightarrow{r}|}, \frac{y}{|\overrightarrow{r}|}, \frac{z}{|\overrightarrow{r}|}.  x \hat{i}, y \hat{j}, z \hat{k} are called vector components
  • If \overrightarrow{r} = x \hat{i} + y \hat{j} + z \hat{k} is a unit vector, then x,y, z are called direction cosines (or) scalar components of \overrightarrow{r}.
  • Zero vector : A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as \overrightarrow{0} .
  • Unit vector : A vector whose magnitude is unity (i.e., 1 unit) is caled a unit vector. The unit vector in the direction of a given vector \overrightarrow{a} is denoted by \hat{a}
  • Coinitial vectors : Two or more vectors having the same initial point are called coinitial vectors.
  • 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.
  • Equal Vectors : Two vectors \overrightarrow{a} and \overrightarrow{b} are said to be equal, if they have the same magnitude and direction regardless of the positions of their points, and written as \overrightarrow{a} = \overrightarrow{b}
  • Free Vectors : vector whose initial points are not specified are called free vectors

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  •  A vector in the direction of \overrightarrow{a} and having magnitude λ units is \lambda \left(\frac{\overrightarrow{a}}{|\overrightarrow{a}|}\right)