bbA=[(3),(1),(-4)] , color(white)(88)bbB=[(4),(-2),(3)]
bb(A*B)
This is the Dot Product and is defined as:
bb(A*B)=||bbA||*||bbB||*costheta
The multiplication of bb(A*B) is different from the normal way in which we multiply in algebra. Normally we multiply in the following way:
(a+b+c)(d+e+f)=ad+ae+af+bd+be+bf+cd+ce+cf
In the dot product we multiply as follows:
(a+b+c)(d+e+f)=ad+be+cf
So we are just multiplying corresponding components and summing the results. This is where the alternative name Inner Product comes from.
For some vector bbA=[(a),(b),(c)]
||bbA|| is the magnitude of the vector bbA and is defined as:
||bbA||=sqrt(a^2+b^2+c^2)
This is just the distance formula found in coordinate geometry.
To the example:
First we find the dot product of bbA and bbB
bb(A*B)=(3xx4)+(1xx-2)+(-4xx3)=-2
Now we find the magnitudes of bbA and bbB
||bbA||=sqrt((3)^2+(1)^2+(-4)^2)=sqrt(9+1+16)=sqrt(26)
||bbB||=sqrt((4)^2+(-2)^2+(3)^2)=sqrt(16+4+9)=sqrt(29)
Now we require:
bb(A*B)-||bbA||||bbB||
-2-sqrt(26)sqrt(29)=color(blue)(-29.46)color(white)(888) (2 .d.p)
