\displaystyle \textbf{1. } \text{If }\overrightarrow a \text{ and }\overrightarrow b\text{ are two non-zero vectors inclined at an angle }\theta,\text{ then:}

\displaystyle \text{(i) }\ \overrightarrow a\cdot\overrightarrow b = |\overrightarrow a|\,|\overrightarrow b|\cos\theta

\displaystyle \text{(ii) }\ \text{Projection of }\overrightarrow a\text{ on }\overrightarrow b = \frac{\overrightarrow a\cdot\overrightarrow b}{|\overrightarrow b|} = \overrightarrow a\cdot\widehat b

\displaystyle \text{(iii) }\ \text{Projection of }\overrightarrow b\text{ on }\overrightarrow a = \frac{\overrightarrow a\cdot\overrightarrow b}{|\overrightarrow a|} = \overrightarrow b\cdot\widehat a

\displaystyle \text{(iv) }\ \text{Projection vector of }\overrightarrow a\text{ on }\overrightarrow b = \left(\frac{\overrightarrow a\cdot\overrightarrow b}{|\overrightarrow b|}\right)\widehat b = \frac{\overrightarrow a\cdot\overrightarrow b}{|\overrightarrow b|^2}\, \overrightarrow b

\displaystyle \text{(v) }\ \text{Projection vector of }\overrightarrow b\text{ on }\overrightarrow a = \left(\frac{\overrightarrow a\cdot\overrightarrow b}{|\overrightarrow a|}\right)\widehat a = \frac{\overrightarrow a\cdot\overrightarrow b}{|\overrightarrow a|^2}\, \overrightarrow a

\displaystyle \text{(vi) }\ \overrightarrow a\cdot\overrightarrow b=0 \iff \overrightarrow a\perp\overrightarrow b

\displaystyle \text{(vii) }\ \overrightarrow a\cdot\overrightarrow b =\overrightarrow b\cdot\overrightarrow a

\displaystyle \text{(viii) }\ \overrightarrow a\cdot\overrightarrow a =|\overrightarrow a|^2

\displaystyle \text{(ix) }\ m(\overrightarrow a\cdot\overrightarrow b) =(m\overrightarrow a)\cdot\overrightarrow b =\overrightarrow a\cdot(m\overrightarrow b)

\displaystyle \text{(x) }\ (m\overrightarrow a)\cdot(n\overrightarrow b) =mn(\overrightarrow a\cdot\overrightarrow b)

\displaystyle \text{(xi) }\ |\overrightarrow a+\overrightarrow b| \le |\overrightarrow a|+|\overrightarrow b|

\displaystyle \text{(xii) }\ |\overrightarrow a-\overrightarrow b| \ge \big||\overrightarrow a|-|\overrightarrow b|\big|

\displaystyle \text{(xiii) }\ |\overrightarrow a+\overrightarrow b|^2 =|\overrightarrow a|^2+|\overrightarrow b|^2 +2(\overrightarrow a\cdot\overrightarrow b)

\displaystyle \text{(xiv) }\ (\overrightarrow a+\overrightarrow b)\cdot (\overrightarrow a-\overrightarrow b) =|\overrightarrow a|^2-|\overrightarrow b|^2

\displaystyle \text{(xv) }\ \overrightarrow a\cdot\overrightarrow b>0 \iff \theta\text{ is acute}

\displaystyle \text{(xvi) }\ \overrightarrow a\cdot\overrightarrow b<0 \iff \theta\text{ is obtuse}

\displaystyle \textbf{2. } \text{If }\overrightarrow a,\overrightarrow b, \overrightarrow c\text{ are three vectors, then}

\displaystyle |\overrightarrow a+\overrightarrow b+\overrightarrow c|^2 =|\overrightarrow a|^2+|\overrightarrow b|^2+|\overrightarrow c|^2 +2(\overrightarrow a\cdot\overrightarrow b +\overrightarrow b\cdot\overrightarrow c +\overrightarrow c\cdot\overrightarrow a)

\displaystyle \textbf{3. } \text{If } \overrightarrow a=a_1\widehat i+a_2\widehat j+a_3\widehat k,\quad \overrightarrow b=b_1\widehat i+b_2\widehat j+b_3\widehat k, \text{ then}

\displaystyle \overrightarrow a\cdot\overrightarrow b =a_1b_1+a_2b_2+a_3b_3

\displaystyle \textbf{4. } \text{If }\overrightarrow a\text{ and } \overrightarrow b\text{ are two vectors inclined at angle }\theta, \text{ then}

\displaystyle \cos\theta =\frac{\overrightarrow a\cdot\overrightarrow b} {|\overrightarrow a|\,|\overrightarrow b|}

\displaystyle \textbf{5. } \text{If }\overrightarrow a,\overrightarrow b, \overrightarrow c\text{ are non-coplanar vectors in space and } \overrightarrow r\text{ is any vector, then}

\displaystyle \overrightarrow r =(\overrightarrow r\cdot\widehat a)\widehat a +(\overrightarrow r\cdot\widehat b)\widehat b +(\overrightarrow r\cdot\widehat c)\widehat c

\displaystyle \text{where }\widehat a,\widehat b,\widehat c \text{ are unit vectors along }\overrightarrow a, \overrightarrow b,\overrightarrow c.

\displaystyle \overrightarrow r =\left(\frac{\overrightarrow r\cdot\overrightarrow a} {|\overrightarrow a|^2}\right)\overrightarrow a +\left(\frac{\overrightarrow r\cdot\overrightarrow b} {|\overrightarrow b|^2}\right)\overrightarrow b +\left(\frac{\overrightarrow r\cdot\overrightarrow c} {|\overrightarrow c|^2}\right)\overrightarrow c

\displaystyle \text{In particular, }\; \overrightarrow r =(\overrightarrow r\cdot\widehat i)\widehat i +(\overrightarrow r\cdot\widehat j)\widehat j +(\overrightarrow r\cdot\widehat k)\widehat k.

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