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

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Let's write to vectors, u and v;

u = xi + yj +zk

v = ai + bj + ck

We'll add u and v:

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.

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**

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