A dot product is an operation that which involves multiplication of two vectors to arrive to a scalar product. Given two vectors, v=ai + bj and u=ci + dj, v.u read as ‘v dot u’ would be equal to a scalar product, ac + bd. So, basically the product would be a number and not a vector.
The dot product of two vectors would be a scalar even in a three dimensional space, R3. So, in a three dimensional space given vectors v=ai +bj+ck and u=xi+yj+zk, the dot-product is given by v.w=ax + by +cz. The definition of dot product can be given as the dot product equation of vectors a’ and b’ such that a.b= ax. bx + ay.by = |a||b|cos(theta) .
Here |a| and |b| are the magnitudes of the vectors and theta is the angle between the vectors. It is read as modulus of vector a multiplied with the modulus of vector b, multiplied by the cosine of the angle between the two vectors a’ and b’.
Following are some of the important points to be remembered while finding the scalar product, i.i=1, j.j=1, k.k=1, i.j=0, j.k=0 and k.i=0, this shows that the scalar product of vectors which are perpendicular to each other is zero.
Some of the properties of dot-product are as given below,
• Commutative property: u.v = v.u
• Distributive property: u.(v+w) = (u.v) + (u.w)
• Associative property: (cv). u = v.(cu)= c(u. v)
• 0. u = u.0 = 0
• v.v =|v|2
• If v. v = 0 then v = 0
Let us now take a look at the dot product proof of distributive property given by u. (v+w)=(u.v)+(u.w)
Let the vectors to be, u=(u_1,u_2,u_3...,u_n ); v =(v_1,v_2,v_3...,v_n) and w=(w_1,w_2,w_3...,w_n). On the left hand side we have, u.(v+w) = (u_1,u_2,u_3...,u_n ).[(v_1,v_2,v_3...,v_n)+ (w_1,w_2,w_3...,w_n)]
= (u_1,u_2,u_3...,u_n ).[(v_1+w_1), (v_2+w_2), (v_3+w_3)…, (v_n+w_n)] on regrouping we get,
= [u_1((v_1+w_1), u_2(v_2+w_2), u_3(v_3+w_3),…,u_n (v_n+w_n)]
Applying the distributive property we get,
= [u_1v_1+u_1w_1, u_2v_2+ u_2w_2, u_3v_3+ u_3w_3….., u_n v_n+ u_n w_n]
Which can be written as, [u_1v_1, u_2v_2, u_3v_3…, u_n v_n] + [u_1w_1, u_2w_2, u_3w_3…, u_n w_n]
On re-writing the above expression we get, [(u_1,u_2,u_3...,u_n ). (v_1,v_2,v_3...,v_n)]+[ (u_1,u_2,u_3...,u_n ). (w_1,w_2,w_3...,w_n)] which would be the expression on the left hand side, [u.v+u.w] and hence proved!Thus we can prove all the properties using the above computational method.
u_n it vectors are the vectors with length of one u_n it. For u_n it vectors u and v, the dot product of u_n it vectors is given by, u.v=cos(theta) where (theta) is the angle between the two u_n it vectors.
