What is the projection of #<8,2,-6 ># onto #<5,-1,7 >#?

1 Answer
Oct 25, 2016

Answer:

The vector projection is #〈-4/15,4/75,-28/75〉#

Explanation:

The vector projection of
#vecb# onto #veca#
is given by
#=(veca.vecb)/(∣veca∣^2)veca#

Let our vectors be #veca=〈a_1,a_2,a_3〉# and #vecb=〈b_1,b_2,b_3〉#

Then the dot product is #veca.vecb=〈a_1,a_2,a_3〉.〈b_1,b_2,b_3〉=a_1b_1+a_2b_2+c_3c_3#

And #∣veca∣^2=〈a_1,a_2,a_3〉.〈a_1,a_2,a_3〉=a_1a_1+a_2a_2+a_3c_3#

#veca=〈5,-1,7〉# and #vecb=〈8,2,-6〉#

Then the dot product is
#veca.vecb=〈5,-1,7〉.〈8,2,-6〉=40-2-42=-4#

and #∣veca∣^2=〈5,-1,7〉〈5,-1,7〉=25+1+49=75#

So the projection is #-4/75〈5,-1,7〉=〈-4/15,4/75,-28/75〉#