Chapter 12 - Vectors and the Geometry of Space - 12.3 The Dot Product - 12.3 Exercises - Page 854: 65

$proj_ab \cdot proj_ba=(a \cdot b) cos^2 \theta$

$proj_ab=\dfrac{a \cdot b}{|a|^2}a=\dfrac{|b| cos \theta }{|a|}a$ and $proj_ba=\dfrac{b \cdot a}{|b|^2}b=\dfrac{|a| cos \theta }{|b|}b$ Therefore, $proj_ab \cdot proj_ba=\dfrac{|b| cos \theta }{|a|}a \cdot \dfrac{|a| cos \theta }{|b|}b$ Hence, $proj_ab \cdot proj_ba=(a \cdot b) cos^2 \theta$