Show that if u is a vector in R^{2} or R&3, then u+(−1)u=0 u+(−1)u=0

Question
Alternate coordinate systems
asked 2021-01-28
Show that if u is a vector in R^{2} or R&3, then \(\displaystyle{u}+{\left(−{1}\right)}{u}={0}\)
\(\displaystyle{u}+{\left(−{1}\right)}{u}={0}\)

Answers (1)

2021-01-29
First, we will show that 0⋅u=0 for any vector u∈R^{2} or R^{3}. Now ​
PSK0\times u+0\times u=(0+0)\times u=0\times u \Rightarrow 0\times u+0\times u+(−0\times u)=0\times u+(−0⋅u) \Rightarrow 0\times u+[0\times u+(−0\times u)]=0\times u+(−0⋅u) \Rightarrow 0\times u+0=0 \Rightarrow 0\times u=0.ZSK ​
Therefore ​
\(\displaystyle{u}+{\left(−{1}\right)}\times{u}={1}\times{u}+{\left(−{1}\right)}\times{u}={\left({1}+{\left(−{1}\right)}\right)}\times{u}{)}={0}\times{u}={0}.\)
PSKu+(−1)\times u =1\times u+(−1)\times u =(1+(−1))\times u) =0\times u =0.ZSK ​
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