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) Ifl = \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.

9) In vector addition

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.