Find a vector which is perpendicular to both vector A and B, where A=2i+3j+4k B=i+2j+3k?

3 Answers
Jul 10, 2017

Answer:

The vector is #=〈1,-2,1〉#

Explanation:

The vector perpendicular to 2 vectors is calculated with the determinant (cross product)

#| (veci,vecj,veck), (d,e,f), (g,h,i) | #

where #〈d,e,f〉# and #〈g,h,i〉# are the 2 vectors

Here, we have #veca=〈2,3,4〉# and #vecb=〈1,2,3〉#

Therefore,

#| (veci,vecj,veck), (2,3,4), (1,2,3) | #

#=veci| (3,4), (2,3) | -vecj| (2,4), (1,3) | +veck| (2,3), (1,2) | #

#=veci(3*3-2*4)-vecj(2*3-1*4)+veck(2*2-3*1)#

#=〈1,-2,1〉=vecc#

Verification by doing 2 dot products

#〈1,-2,1〉.〈2,3,4〉=1*2-2*+1*4=0#

#〈1,-2,1〉.〈1,2,3〉=1*1-2*2+1*3=0#

So,

#vecc# is perpendicular to #veca# and #vecb#

Jul 10, 2017

Answer:

#i-2j+k#

Explanation:

Cross product of vectors A and B is perpendicular to each vector A and B.

#vecA#x #vecB=i(9-8)-j(6-4)+k(4-3)#

#vecA#x #vecB=i-2j+k#

Jul 10, 2017

We know that cross product of any two vectors yields a vector which is perpendicular to both vectors
#:.# for two vectors #vecA and vecB# if #vecC# is the vector perpendicular to both.
#vecC=vecAxxvecB=##[(hati, hatj, hatk), (A_1, A_2,A_3),(B_1, B_2, B_3)]#
#=(A_2B_3−B_2A_3)hati−(A_1B_3−B_1A_3)hatj+(A_1B_2−B_1A_2)hatk#.
Inserting given vectors we obtain
#vecC=##[(hati, hatj, hatk), (2, 3,4),(1, 2, 3)]#
#=(3xx3-2xx4)hati−(2xx3−1xx4)hatj+(2xx2−1xx3)hatk#.
#=hati−2hatj+hatk#.