Position Vector Of A Point And Unit Vector:

• Position Vector Of A Point

$\quad \quad$ let $O$ be a fixed origin, then the position vector of a point $P$ is the vector $\overrightarrow{O P}$. If $\vec{a}$ and $\vec{b}$ are position vectors of two points $A$ and $B$, then, $\overrightarrow{A B}=\vec{b}-\vec{a}$

• Unit vector $\quad$ $$\hat{n}=\frac{\text { Vector }}{\text { Its modulus }}=\frac{\vec{a}}{|\vec{a}|}$$

Properties Of Addition:

• $ \vec{a}+\vec{b}=\vec{b}+\vec{a}$ $\quad $(Commutative)

• $ \vec{a}+(\vec{b}+\vec{c})=(\vec{a}+\vec{b})+\vec{c}$ $\quad $(Associativity )

• $ \vec{a}+\overrightarrow{0}=\vec{a}$ $\quad $(Additive identity)

• $ \vec{a}+(-\vec{a})=\overrightarrow{0}$ $\quad $(Additive inverse)

• $ c(\vec{a}+\vec{b})=c \vec{a}+c \vec{b}$

• $ (c+d) \vec{a}=c \vec{a}+d \vec{a}$

• $ (c d) \vec{a}=c(d \vec{a})$

• $ 1 \times \vec{a}=\vec{a}$

Distance Formula:

$\quad$ Distance between the two points $A(\vec{a})$ and $B(\vec{b})$ is $A B=|\vec{a}-\vec{b}|$

Section Formula:

$$\overrightarrow{\mathrm{r}}=\frac{\mathrm{n} \overrightarrow{\mathrm{a}}+\mathrm{m} \overrightarrow{\mathrm{b}}}{\mathrm{m}+\mathrm{n}}$$

$\quad$ Mid point of $A B=\frac{\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}}{2}$.

Scalar Product Of Two Vectors:

PYQ-2023-Vector_Algebra-Q1, PYQ-2023-Vector_Algebra-Q4, PYQ-2023-Vector_Algebra-Q5, PYQ-2023-Vector_Algebra-Q7, PYQ-2023-Vector_Algebra-Q8, PYQ-2023-Vector_Algebra-Q10, PYQ-2023-Vector_Algebra-Q11, PYQ-2023-Vector_Algebra-Q12, PYQ-2023-Vector_Algebra-Q13, PYQ-2023-Vector_Algebra-Q14, PYQ-2023-Vector_Algebra-Q15, PYQ-2023-Vector_Algebra-Q17, PYQ-2023-Vector_Algebra-Q20, PYQ-2023-Vector_Algebra-Q22, PYQ-2023-Vector_Algebra-Q23, PYQ-2023-3D-Q13

$$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos \theta$$

$\quad$ where $|\vec{a}|$ and $|\vec{b}|$ are the magnitudes of $\vec{a}$ and $\vec{b}$ respectively, and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

• $ i . i=j . j=k . k=1 ; \quad i . j=j . k=k . i=0 $

• projection of $\vec{a}$ on $\vec{b}=\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

• If $\vec{a}=a_{1} i+a_{2} j+a_{3} k $ & $\vec{b}=b_{1} i+b_{2} j+b_{3} k$ then $\vec{a} \cdot \vec{b}=a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}$

• The angle $\phi$ between $\vec{a}$ & $\vec{b}$ is given by $\cos \phi=\frac{\vec{a} \vec{b}}{|\vec{a}||\vec{b}|}, 0 \leq \phi \leq \pi$

• $ \vec{a} \cdot \vec{b}=0 \Leftrightarrow \vec{a} \perp \vec{b} \quad(\vec{a} \neq 0 \vec{b} \neq 0)$

Vector Product Of Two Vectors:

PYQ-2023-Vector_Algebra-Q1, PYQ-2023-Vector_Algebra-Q2, PYQ-2023-Vector_Algebra-Q3, PYQ-2023-Vector_Algebra-Q5, PYQ-2023-Vector_Algebra-Q7, PYQ-2023-Vector_Algebra-Q10, PYQ-2023-Vector_Algebra-Q11, PYQ-2023-Vector_Algebra-Q12, PYQ-2023-Vector_Algebra-Q13, PYQ-2023-Vector_Algebra-Q14, PYQ-2023-Vector_Algebra-Q17, PYQ-2023-Vector_Algebra-Q19, PYQ-2023-Vector_Algebra-Q21, PYQ-2023-Vector_Algebra-Q23, PYQ-2023-3D-Q23, PYQ-2023-3D-Q27

• If $\vec{a} , \vec{b}$ are two vectors & $\theta$ is the angle between them then

$$\vec{a} \times \vec{b} = |\vec{a}||\vec{b}| \sin \theta \vec{n}$$

$\quad$ where $\vec{n}$ is the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{a}$, $\vec{b}$, and $\vec{n}$ form a right-handed screw system.

• Geometrically $|\vec{a} \times \vec{b}|=$ area of the parallelogram whose two adjacent sides are represented by $\vec{a} $&$ \vec{b}$.

• $ \hat{i} \times \hat{i}=\hat{j} \times \hat{j}=\hat{k} \times \hat{k}=\overrightarrow{0} ; \hat{i} \times \hat{j}=\hat{k}, \hat{j} \times \hat{k}=\hat{i}, \hat{k} \times \hat{i}=\hat{j}$

• If $\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k} \quad $ & $ \vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}$ then $\vec{a} \times \vec{b}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3}\end{array}\right|$

• $ \vec{a} \times \vec{b}=\overrightarrow{0} \Leftrightarrow \vec{a}$ and $\vec{b}$ are parallel (collinear) $(\vec{a} \neq 0, \vec{b} \neq 0)$ i.e. $\vec{a}=K \vec{b}$, where $K$ is a scalar.

• Unit vector perpendicular to the plane of $\vec{a} $& $\vec{b}$ is $\hat{n}= \pm \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$

• If $\vec{a}, \vec{b}$ & $\vec{c}$ are the pv’s of 3 points $A, B $& $ C$ then the vector area of triangle $A B C=\frac{1}{2}[\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}]$

• The points $A, B $ & $ C$ are collinear if $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}$

• Area of any quadrilateral whose diagonal vectors are $\overrightarrow{d}_1 $ & $ \overrightarrow{d}_2 $ is given by $\frac{1}{2}|\overrightarrow{d}_1 \times \overrightarrow{d}_2|$

Lagrange’s Identity:

PYQ-2023-Vector_Algebra-Q16, PYQ-2023-Vector_Algebra-Q19

$$(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})^{2}=|\overrightarrow{\mathrm{a}}|^{2}|\overrightarrow{\mathrm{b}}|^{2}-(\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}})^{2}=|\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \quad \overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}}|$$

Scalar Triple Product:

PYQ-2023-Vector_Algebra-Q6, PYQ-2023-Vector_Algebra-Q8, PYQ-2023-Vector_Algebra-Q9, PYQ-2023-Vector_Algebra-Q18, PYQ-2023-Vector_Algebra-Q20, PYQ-2023-Function-Q10

• The scalar triple product of three vectors $\vec{a}$, $\vec{b}$ $\vec{c}$ is defined as $\vec{a} \times \vec{b} \cdot \vec{c}=|\vec{a}||\vec{b}||\vec{c}| \sin \theta \cos \phi$

• Volume of tetrahydron $V=[\vec{a} \bar{b} \vec{c}]$

• In a scalar triple product the position of dot & cross can be interchanged i.e.

• $ \vec{a} \cdot(\vec{b} \times \vec{c})=(\vec{a} \times \vec{b}) \cdot \vec{c} \quad$ OR $[\vec{a} \vec{b} \vec{c}]=[\vec{b} \vec{c} \vec{a}]=[\vec{c} \vec{a} \vec{b}]$

• $ \vec{a} \cdot(\vec{b} \times \vec{c})=-\vec{a} \cdot(\vec{c} \times \vec{b})$ i.e. $[\vec{a} \vec{b} \vec{c}]=-[\vec{a} \vec{c} \vec{b}]$

• If $\vec{a}=a_{1} i+a_{2} j+a_{3} k ; \vec{b}=b_{1} i+b_{2} j+b_{3} k $ and $\vec{c}=c_{1} i+c_{2} j+c_{3} k$ then $[\vec{a} \vec{b} \vec{c}]=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right|$

• If $\vec{a}, \vec{b}, \vec{c}$ are coplanar $\Leftrightarrow[\vec{a} \vec{b} \vec{c}]=0$.

• Volume of tetrahedron $O A B C$ with $O$ as origin & $ A(\vec{a}), B(\vec{b})$ and $C(\vec{c})$ be the vertices $=\left|\frac{1}{6}[\vec{a} \vec{b} \vec{c}]\right|$

• The positon vector of the centroid of a tetrahedron if the pv’s of its vertices are $\vec{a}, \vec{b}, \vec{c}$ & $\vec{d}$ are given by $\frac{1}{4}[\vec{a}+\vec{b}+\vec{c}+\vec{d}]$.

Vector Triple Product:

• $ \vec{a} \times(\vec{b} \times \vec{c})=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}$

• $ (\vec{a} \times \vec{b}) \times \vec{c}=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{b} \cdot \vec{c}) \vec{a}$

• $(\vec{a} \times \vec{b}) \times \vec{c} \neq \vec{a} \times(\vec{b} \times \vec{c})$

Reciprocal System Of Vectors:

$\quad$ If $\vec{a}, \vec{b}, \vec{c} \hspace{1mm}$ & $\hspace{1mm} \vec{a}^{\prime}, \vec{b}^{\prime}, \vec{c}^{\prime}$ are two sets of non coplanar vectors such that $\vec{a} \cdot \vec{a}^{\prime}=\vec{b} \cdot \vec{b}^{\prime}=\vec{c} \cdot \vec{c}^{\prime}=1$ then the two systems are called Reciprocal System of vectors, where

$$\vec{a}^{\prime}=\frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{b}^{\prime}=\frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \vec{c}^{\prime}=\frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$$