vector problemExplain why when add two vectors the result is a vector but when multiply vectors, the result is  number?

santaone | High School Teacher | (Level 3) eNoter

Posted on

That is not true, really.

Vectors can be added graphically or by resolving them into components along co-ordinate axes.

as for  r = x1 i + y1 j + z1 k ( bold indicates vector )

and s = x2 i + y2 j + z2 k

r + s  = (x1 + x2) i +(y1 +y2)j+(z1+z2)k

Vectors have two kinds of products, Dot product which is a scalar and a cross product which is a vector.

Dot product is given by

r . s = x1*x2 + y1*y2 +z1*z2

and cross product is given by

r X s = (y1*z2-y2*z1) i - (x1*z2 - x2*z1) j + (x1*y2-x2*y1)k

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Let's write to vectors, u and v;

u = xi + yj +zk

v = ai + bj + ck

u+v = (xi + yj +zk) + (ai + bj + ck)

We'll remove the brackets and combine like terms:

u+v = (xi + ai) +  (yj + bj) + (zk + ck)

u + v = i(x +a) + j(y + b) + k(z+c)

Since the versors i,j,k exist still, the result of addition of two vectors is a vector, also.

Let's multiply u and v:

u*v = (xi + yj +ck)*(ai + bj + zk)

u*v = x*a*i^2 + x*b*i*j + .... + y*b*j^2 + .... + c*z*k^2

Since the dot products of the versors i*i = i^2 ; j*j = j^2 ; k*k = k^2 is 1 and the dot product of the versors i*j ; j*k ; i*k is zero, we'll get:

u*v = ax + by + cz

We notice that the result of addition of two vectors is a vector and the result of dot product of two vectors is a number.