# 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

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
1. Geometrical interpretation of scalar product:

=

â‡’ = projection length of vector in the direction of

Similarly, projection length of vector in the direction of is
2. â‡’
3. (i.e., commutative)
4. (distributive)
5. (where l and m are scalars)
6. are perpendicular to each other
7. =
8. If then = a1b1 + a2b2 + a3b3
9. If are non-zero, then the angle between them is given by
10. 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
1. Two non- zero vectors are collinear if and only if
2. If , then = (a2b3 â€“ a3b2)
3.

â‡’

= 2 (Area of triangle AOC)

= Area of parallelogram.

The area of the triangle OAB is .
4. The unit vectors perpendicular to the plane of
5. If are diagonals of parallelogram; then its

Area =

# Scalar triple product

The scalar triple product of three vectors is defined as

We denote it by .

Properties of scalar triple product
1.
i.e.,

=
2.
then

Here represents (and is equal to) the volume of the parallelepiped whose adjacent sides are represented by the vectors Three vectors are coplanar if = 0
3. =
4. , i.e., position of dot and cross can be interchanged without altering the order. Hence it is also represented by
5. (where k is a scalar)
6. i.e., if any two vectors are same then vectors are coplanar
7. 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 .