How (7i+3j-5k) *(i-j+k)=(2i+12j+10k?

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Answer 1 · 2018-02-02T17:33:15

Vector cross means,

#vec A × vec B = |vec A| |vec B| sin theta#(where #theta# is the angle between the two vectors)

Rule of vector cross,

#i×i=0# (as #theta=0#,so #sin theta=0#) remember #i# means an unit vector along #X# axis, #j# means along #Y# and #k# along #Z# axis)

#j×j=0# ( do )
#k×k=0# ( do )

Now #i×j = k , j×k= i# and #k×i = j# (as #theta =90# so #sin theta =1#)

in the opposite direction,

#j×i=-k , k×j = -i# and #i×k = -j#

So, #(i-j+k)×(7i+3j-5k) = (3k+5j+7k+5i+7j-3i)=(2i+12j+10k)#

Answer 2 · 2018-02-02T18:55:04

This is a cross product

Perform the dot product of

#<7,3,-5> . <2, 12, 10> = ((7)*(2)+(3)*(12)+(-5)*(10))=14+36-50=0#

As the dot product is #=0#, the vectors #<7,3,-5># and #<2, 12, 10> # are orthogonal.

Also,

#<1,-1,1> . <2, 12, 10> = ((1)*(2)+(-1)*(12)+(1)*(10))=2-12+10=0#

As the dot product is #=0#, the vectors #<1,-1,1># and #<2, 12, 10> # are orthogonal.

As the vector #<2, 12, 10># is perpendicular to both #<7,3,-5># and #<1,-1,1>#, we think that there is a cross product.

#|(hati,hatj,hatk),(1,-1,1),(7,3,-5)| =hati*|(-1,1),(3,-5)|-hatj*|(1,1),(7,-5)|+hatk*|(1,-1),(7,3)|#

#=hati*(5-3)-hatj*(-5-7)+hatk*(3+7)#

#= <2, 12,10>#