please send all Formulas used in vectors class11
Answers
The position vector of any point p(x,y) is
op = \dbinom{x}{y} or OP = ( x,y ).
2.The magnitude of position vector
OP = \sqrt{x^2 + y^2} and direction \tan \theta = \dfrac{y}{x}
3. The unit vector = \dbinom{1}{0} where the magnitude of unit vector is 1
Or,the unit vector = \dfrac{vector}{its modulus} = \dfrac{ \overrightarrow{a}}{ |\overrightarrow{a}| }
4.The two vectors \overrightarrow{a} and \overrightarrow{b} are parallel if \overrightarrow{a} = k \overrightarrow{b} and \overrightarrow{b} = m \overrightarrow{a} where k and m are the scalars.
5.If \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} then \overrightarrow{AC} is the result vector which is the triangle law of vector addition.
6. The scalar or dot product of any two vectors \overrightarrow{a} . \overrightarrow{b} = | \overrightarrow{a} | | \overrightarrow{b} | cos\theta.
7. The angle between two vectors is \cos \theta = \dfrac{\overrightarrow{a} . \overrightarrow{b}} {| \overrightarrow{a} | | \overrightarrow{b} |}
8. \overrightarrow{a} = x_1 i + y_1 j and \overrightarrow{b} = x_2 i + y_2 j , then :
\overrightarrow{a} . \overrightarrow{b} = x_1 . x_2 + y_1 . y_2 where i.j = j.1 = 0
9. If the position vector of A is \overrightarrow{a} , position vector of point B is \overrightarrow{b} and position vector of mid-point M is m then \overrightarrow{m} = \dfrac{\overrightarrow{a} + \overrightarrow{b}}{2}
10. If the point P divides Ab internally in the ratio m:n then position vector of P is given by \overrightarrow{p} = \dfrac{n \overrightarrow{a} + m \overrightarrow{b}}{m + n} which is a section formula.
11.If P divides AB externally in the ratio m:n then \overrightarrow{p} = \dfrac{m \overrightarrow{b} - n \overrightarrow{a}}{m - n}
PRODUCT OF TWO VECTORS
1.Scalar Product ( dot product )
Let \overrightarrow{a} = (a_1,a_2) , \overrightarrow{b} = (b_1,b_2) then dot product of \overrightarrow{a} & \overrightarrow{b} is devoted by \overrightarrow{a}.\overrightarrow{b} read as \overrightarrow{a} dot \overrightarrow{b} and defined by \overrightarrow{a}.\overrightarrow{b} = a_1 b_1 , a_2 b_2
Note:
if\overrightarrow{a} = (a_1,a_2,a_3),\overrightarrow{b} = (b_1,b_2,b_3) \\ \overrightarrow{a} . \overrightarrow{b} = a_1b_1 + a_2b_2 +a_3b_3
OR
The scalar product of \overrightarrow{a} & \overrightarrow{b} is devoted by \overrightarrow{a} . \overrightarrow{b} ,
\overrightarrow{a} . \overrightarrow{b} |\overrightarrow{a}|.|\overrightarrow{b}| \cos \theta where \theta being angle between \overrightarrow{a} & \overrightarrow{b}
Note:1
\cos \theta = \dfrac{\overrightarrow{a} . \overrightarrow{b}}{| \overrightarrow{a} | | \overrightarrow{b} |}
Note:2
\overrightarrow{a} & \overrightarrow{b} are perpendicular if \theta = 90^o
i.e \overrightarrow{a} . \overrightarrow{b} |\overrightarrow{a}|.|\overrightarrow{b}| \cos 90^o or \overrightarrow{a} . \overrightarrow{b} = 0
2.Properties of Scalar Product
i. \overrightarrow{a} . \overrightarrow{b} = \overrightarrow{b} . \overrightarrow{a}.
ii. m \overrightarrow{a} . n \overrightarrow{b} - mn \overrightarrow{a}. \overrightarrow{b} = \overrightarrow{a} mn \overrightarrow{b}.
iii. \overrightarrow{a}( \overrightarrow{b} + \overrightarrow{c} ) = \overrightarrow{a} . \overrightarrow{c}
iv. ( \overrightarrow{b} + \overrightarrow{c} )^2 = \overrightarrow{a}^2 + 2.\overrightarrow{a}.\overrightarrow{b} + \overrightarrow{b}^2
v. If \overrightarrow{i} = (1,0,0): \overrightarrow{j} = (0,1,0), \overrightarrow{k} = (0,0,1)then \overrightarrow{i} . \overrightarrow{j} = \overrightarrow{j} . \overrightarrow{k} = \overrightarrow{k} . \overrightarrow{i} = \overrightarrow{i} . \overrightarrow{k} = \overrightarrow{j} . \overrightarrow{k} = 0
3.Vector (cross) Product of two vectors.
Let \overrightarrow{a} = (a_1 , a_2 , a_3 ), \overrightarrow{b} = (b_1 , b_2 , b_3 ) be two vectors then the cross product of \overrightarrow{a} \times \overrightarrow{b}is devoted by\overrightarrow{a} \times \overrightarrow{b} and defined by
\overrightarrow{a} \times \overrightarrow{b} = (a_1 , a_2 , a_3 ) \times (b_1 , b_2 , b_3 )
= \begin{pmatrix} a_1 & a_2 & a_3 & a_1 & a_2 \\ b_1 & b_2 & b_3 & b_1 & b_2 \end{pmatrix} = ( a_2 b_3 - a_3 b_2 , a_3 b_1 - a_1 b_2 - a_2 b_1 )