How do you find the scalar and vector projections of b onto a given #a = i + j + k#, #b = i - j + k#?

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dk_ch Share
Jan 30, 2018

Answer:

Saclar projection#1/sqrt3# and Vector projection #1/3(hati+hatj+hatk)#

Explanation:

We have been given two vectors #veca and vec b# , we are to find out the scalar and vector projection of #vec b # onto #vec a#
we have #veca=hati+hatj+hatk# and #vecb=hati-hatj+hatk#
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The scalar projection of #vec b # onto #vec a# means the magnitude of resolved component of #vec b # inthe direction of #vec a# and is given by
The scalar projection of #vec b # onto #vec a# #=(vecb*veca)/|veca|#
#=((hati+hatj+hatk)*(hati-hatj+hatk))/(sqrt(1^2+1^1+1^2))#
#=(1^2-1^2+1^2)/sqrt3=1/sqrt3#

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The Vector projection of #vec b # onto #vec a# means the resolved component of #vec b # in the direction of #vec a# and is given by
The vector projection of #vec b # onto #vec a# #=(vecb*veca)/|veca|^2.(hati+hatj+hatk)#
#=((hati+hatj+hatk)*(hati-hatj+hatk))/((sqrt(1^2+1^1+1^2))^2).(hati+hatj+hatk)#
#=(1^2-1^2+1^2)/3.(hati+hatj+hatk)=1/3(hati+hatj+hatk)#

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