Question

if A=2i+j+k, B=i-2j+2k and C=3i-4j+2k , find the projection A+C in the direction of B.

Select one:

A 173

B 203

C 143

D 193

Answer 1

$A+C=\left( 5,-3,3 \right) \\B=\left( 1,-2,2 \right) \\proj_B\left( A+C \right) =\frac{\left( B,A+C \right)}{\left\| B \right\| ^2}B=\frac{5\cdot 1+\left( -3 \right) \cdot \left( -2 \right) +3\cdot 2}{1^2+2^2+2^2}\left( 1,-2,2 \right) =\\=\frac{17}{9}\left( 1,-2,2 \right) \\\left| proj_B\left( A+C \right) \right|=\frac{17}{9}\sqrt{1^2+\left( -2 \right) ^2+2^2}=\frac{17}{3}\\None\,\,is\,\,correct$