#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)
