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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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