## Dot Product

### Scalar or Dot product

**(1) Scalar or Dot product of two vectors:**

If a and b are two non-zero vectors and θ be the angle between them, then their scalar product (or dot product) is denoted by **a.b** and is defined as the scalar |a||b| cos θ , where |a| and |b| are modulii of a and b respectively and 0≤ θ≤π. Dot product of two vectors is a scalar quantity.

**Angle between two vectors:**

If a, b be two vectors inclined at an angle θ, then a.b = |a||b| cos θ.

**(2) Properties of scalar product:**

**Commutativity:**The scalar product of two vector is commutative i.e., a . b = b . a.**Distributivity of scalar product over vector addition:**The scalar product of vectors is distributive over vector addition i.e.,

(a) a.(b + c) = a . b − a . c, (Left distrihutivity)

(b) (b + c).a = b . a + c . a , (Right distrihutivity)- Let a and b be two non-zero vectors a . b = 0 ⇔ a ⊥ b.

As i, j, k are mutually perpendicular unit vectors along the co-ordinate axes, therefore, i . j = j . i =0 ; j . k = k . j = 0; k . i = i . k = 0. - For any vector a, a . a = |a|
^{2}.

As i, j, k are unit vectors along the co-ordinate axes, therefore i . i = |i|^{2}, j . j = |j|^{2}and k . k = |k|^{2} - If m, n are scalars and a, b be two vectors, then ma . nb = mn(a . b) = (mn a).b = a.(mn b)
- For any vectors a and b, we have

(a) a. (−b) = − (a.b) = (−a).b

(b) (−a).( −b) = a.b - For any two vectors a and b, we have
- |a + b|
^{2}= |a|^{2}+ |b|^{2}+ 2a.b - |a − b|
^{2}= |a|^{2}+ |b|^{2}+ 2a.b - (a+b).(a—b) = |a|
^{2}− |b|^{2} - |a + b| = |a| + |b| ⇒ a ∥ b
- |a + b|
^{2}= |a|^{2}+ |b|^{2}⇒ a ⊥ b - |a + b| = |a − b| = ⇒ a ⊥ b

- |a + b|

**(3) Scalar product in terms of components:**

If a = a_{1}i + a_{2}j + a_{3}k and b = b_{1}i + b_{2}j + b_{3}k then a . b = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}.

The components of b along and perpendicular to are and respectively.

**(4) Work done by a force:**

If a number of forces are acting on a particle, then the sum of the works done by the separate forces is equal to the work done by the resultant force.

### Scalar triple product

**(1) Scalar triple product of three vectors:**

If a, b, c are three vectors, then their scalar triple product is defined as the dot product of two vectors a and b × c. It is generally denoted by a . (b × c) or [a b c].

**(2) Properties of scalar triple product**:

- If a, b, c are cyclically permuted, the value of scalar triple product remains the same. i.e., (a × b). c = (b × c). a = (c × a). b or [a b c] = [b c a] = [c a b]
- The change of cyclic order of vectors in scalar triple product changes the sign of the scalat’ triple product but not the magnitude i.e., [a b c] = −[b a c] = −[c b a] = −[a c b]
- In scalar triple product the positions of dot and cross can be interchanged provided that the cyclic order of the vectors remains same i.e., (a × b). c = a. (b × c)
- The scalar triple product of three vectors is zero if any two of them are equal.
- For any three vectors a, b, c and scalar λ, [λ a b c] = λ[a b c]
- The scalar triple product of three vectors is zero if any two of them are parallel or collinear.
- If a, b, c, d are four vectors, then [(a + b) c d] = [a c d] + [b c d].
- The necessary and sufficient condition for three non-zero non-collinear vectors to be coplanar is that [a b c] = 0.
- Four points with position vectors a, b, c and d will be coplanar, if [ a b c] + [d c a] + [d a b] = [a b c].
- Volume of parallelopiped whose coterminous edges are a, b, c is [a b c] or a(b × c).

**(3) Scalar triple product in terms of components:**

**(4) Tetrahedron:**

A tetrahedron is a three-dimensional figure formed by four triangle OABC is a tetrahedron with ∆ABC as the base. OA, OB, OC, AB, BC and CA are known as edges of the tetrahedron. OA, BC; OB, CA and OC, AB are known as the pairs of opposite edges. A tetrahedron in which all edges are equal, is called a regular tetrahedron. Any two edges of regular tetrahedron are perpendicular to each other.

**Volume of tetrahedron**

- The volume of a tetrahedron
- If a, b, c are position vectors of vertices A, B and C with respect to O, then volume of tetrahedron OABC = \(\frac { 1 }{ 6 }\) [a b c].
- If a, b, c, d are position vectors of vertices A, B, C, D of a tetrahedron ABCD, then its volume = \(\frac { 1 }{ 6 }\) [b−a c−a d−a].

**(5) Reciprocal system of vectors:**

Let be three non-coplanar vectors, and let . a*‘*, b*‘*, c*‘* are said to form a reciprocal system of vectors for the vectors a, b, c.

If a, b, c and a*‘*, b*‘*, c*‘* form a reciprocal system of vectors, then

### Scalar product of four vectors

(a × b) . (c × d) is a scalar product of four vectors. It is the dot product of the vectors a × b and c × d.

It is a scalar triple product of the vectors a,b and c × d as well as scalar triple product of the vectors a × b, c and d.