# Basic Formulas and Results of Vectors

1) If $\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$ then the magnitude or length or norm or absolute value of $\overrightarrow a$ is $\left| {\overrightarrow a } \right| = a = \sqrt {{x^2} + {y^2} + {z^2}}$

2) A vector of unit magnitude is the unit vector. If $\overrightarrow a$ is a vector then the unit vector of $\overrightarrow a$ is denoted by $\widehat a$ and $\widehat a = \frac{{\overrightarrow a }}{{\left| {\overrightarrow a } \right|}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\therefore \overrightarrow a = \left| {\overrightarrow a } \right|\widehat a$

3) Important unit vectors are $\widehat i,\widehat j,\widehat k$, where $\widehat i = \left[ {1,0,0} \right],\widehat j = \left[ {0,1,0} \right],\widehat k = \left[ {0,0,1} \right]$

4) The components of unit vectors of $\overrightarrow a$ are called the direction cosines of $\overrightarrow a$, denoted by $l,m,n$ and ${l^2} + {m^2} + {n^2} = 1$

5) If$l = \cos \alpha ,\,m = \cos \beta ,\,n = \cos \gamma$, then $\alpha ,\beta ,\gamma$ are called directional angles of the vectors $\overrightarrow a$ and ${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$

6) If $A\left( {{a_1},{a_2},{a_3}} \right)$ and $B\left( {{b_1},{b_2},{b_3}} \right)$ are the two points, then $\overrightarrow {AB} = \left( {{b_1} - {a_1},{b_2} - {a_2},{b_3} - {a_3}} \right)$

7) If $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$ and $\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + {b_3}\widehat k$ then
$\overrightarrow a + \overrightarrow b = \left( {{a_1} + {b_1}} \right)\widehat i + \left( {{a_2} + {b_2}} \right)\widehat j + \left( {{a_3} + {b_3}} \right)\widehat k$
$\overrightarrow a - \overrightarrow b = \left( {{a_1} - {b_1}} \right)\widehat i + \left( {{a_2} - {b_2}} \right)\widehat j + \left( {{a_3} - {b_3}} \right)\widehat k$

8) If $\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$ and $\lambda$ is a scalar number, then$\lambda \overrightarrow a = \lambda {a_1}\widehat i + \lambda {a_2}\widehat j + \lambda {a_3}\widehat k$.

I. $\overrightarrow a + \overrightarrow b = \overrightarrow b + \overrightarrow a$
II. $\overrightarrow a + \left( {\overrightarrow b + \overrightarrow c } \right) = \left( {\overrightarrow a + \overrightarrow b } \right) + \overrightarrow c$
III. $k\left( {\overrightarrow a + \overrightarrow b } \right) = k\overrightarrow a + k\overrightarrow b$
IV. $\overrightarrow a + \overrightarrow 0 = \overrightarrow 0 + \overrightarrow a$, $\therefore \overrightarrow 0$is the additive identity in vector addition.
V. $\overrightarrow a + \left( { - \overrightarrow a } \right) = - \overrightarrow a + \overrightarrow a = \overrightarrow 0$, $\therefore \overrightarrow a$is the inverse in vector addition.