Section Formula
Internal division Let A and B be two points with position vectors respectively, and C be a point dividing AB internally in the ratio m : n. Then the position vector of C is given by
External division Let A and B be two points with position vectors respectively and let C be a point dividing externally in the ratio m : n. Then the position vector of is given by
Addition of two vectors
Let and be any two vectors.
Then their sum or resultant, denoted by , is defined as vector given by the diagonal of the parallelogram OACB, as shown in the figure, i.e., .
Scalar product of two vectors
Definition = cos Î¸, where Î¸ is the angle between the vectors, where 0 â‰¤ Î¸ â‰¤ Ï€.
For Î¸ acute, obtuse, and right angle is positive, negative, and zero respectively.
Properties of scalar product
 Geometrical interpretation of scalar product:
 â‡’
 (i.e., commutative)
 (distributive)
 (where l and m are scalars)
 are perpendicular to each other
 =
 If then = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}
 If are nonzero, then the angle between them is given by
 Let , Taking dot product with and alternatively, we have x = . Then
Vector (or cross) product of two vectors
Definition = a b sin Î¸ , where Î¸ is the angle between , (0 â‰¤ Î¸ â‰¤ Ï€), and is a unit vector along the line perpendicular to both .
Properties of vector product
 Two non zero vectors are collinear if and only if
 If , then = (a_{2}b_{3} â€“ a_{3}b_{2})

 The unit vectors perpendicular to the plane of
 If are diagonals of parallelogram; then its
Scalar triple product
The scalar triple product of three vectors is defined as
We denote it by .
Properties of scalar triple product


 =
 , i.e., position of dot and cross can be interchanged without altering the order. Hence it is also represented by
 (where k is a scalar)
 i.e., if any two vectors are same then vectors are coplanar
 Four points with position vectors and are coplanar if
Vector triple product
The vector triple product of three vectors is the vector:
also
In general,
If then the vector are collinear.
is a vector perpendicular to but is vector perpendicular to the plane of .
â‡’ the vector must lie in plane of
Note:
Also
which shows that vector lies in the plane of and also in the plane of vectors .
Thus the vector lies along the common section of the plane of and the plane of .