To prove: The formula ( u → × v → ) × w → = ( u → ⋅ w → ) v → − ( v → ⋅ w → ) u → and u → × ( v → × w → ) = ( u → ⋅ w → ) v → − ( u → ⋅ v → ) w → for the vectors u → = 2 i → , v → = 2 j → , w → = 2 k → by evaluating both the sides and comparing the results.
To prove: The formula ( u → × v → ) × w → = ( u → ⋅ w → ) v → − ( v → ⋅ w → ) u → and u → × ( v → × w → ) = ( u → ⋅ w → ) v → − ( u → ⋅ v → ) w → for the vectors u → = 2 i → , v → = 2 j → , w → = 2 k → by evaluating both the sides and comparing the results.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Chapter 11, Problem 13AAE
(a)
To determine
To prove: The formula (u→×v→)×w→=(u→⋅w→)v→−(v→⋅w→)u→ and u→×(v→×w→)=(u→⋅w→)v→−(u→⋅v→)w→ for the vectorsu→=2i→,v→=2j→,w→=2k→ by evaluating both the sides and comparing the results.
(b)
To determine
To prove: The formula (u→×v→)×w→=(u→⋅w→)v→−(v→⋅w→)u→ and u→×(v→×w→)=(u→⋅w→)v→−(u→⋅v→)w→ for the vectors u→=i→−j→+k→,v→=2i→+j→−2k→,w→=−i→+2j→−k→ by evaluating both the sides and comparing the results.
(c)
To determine
To prove: The formula (u→×v→)×w→=(u→⋅w→)v→−(v→⋅w→)u→ and u→×(v→×w→)=(u→⋅w→)v→−(u→⋅v→)w→ for the vectors u→=2i→+j→,v→=2i→−j→+k→,w→=i→+2k→ by evaluating both the sides and comparing the results.
(d)
To determine
To prove: The formula (u→×v→)×w→=(u→⋅w→)v→−(v→⋅w→)u→ and u→×(v→×w→)=(u→⋅w→)v→−(u→⋅v→)w→ for the vectors u→=i→+j→−2k→,v→=−i→−k→,w→=2i→+4j→−2k→ by evaluating both the sides and comparing the results.