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$$\lambda \overrightarrow a = \lambda {a_1}\widehat i + \lambda {a_2}\widehat j + \lambda {a_3}\widehat k$$.

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.