Chapter 7 Problem 18

Problem

Calculate the angle between the vectors: ${\vec {A}}=6.8{\hat {i}}-3.4{\hat {j}}-6.2{\hat {k}}$ and ${\vec {B}}=8.2{\hat {i}}+2.3{\hat {j}}-7.0{\hat {k}}$

Solution

${\vec {A}}\cdot {\vec {B}}=(6.8)(8.2){\hat {i}}.{\hat {i}}+(-3.4)(2.3){\hat {j}}.{\hat {j}}+(-6.2)(-7.0){\hat {k}}.{\hat {k}}=91.34$

$A={\sqrt {{\vec {A}}\cdot {\vec {A}}}}={\sqrt {(6.8)^{2}+(-3.4)^{2}+(-6.2)^{2}}}=9.81$

$B={\sqrt {{\vec {B}}\cdot {\vec {B}}}}={\sqrt {(8.2)^{2}+(2.3)^{2}+(-7.0)^{2}}}=11.0$

${\vec {A}}\cdot {\vec {B}}=AB\cos \theta \rightarrow \theta =\cos ^{-1}{\frac {{\vec {A}}\cdot {\vec {B}}}{AB}}={\frac {91.34}{(9.81)(11.0)}}=32^{\circ }$