Must show all your work!!! Verify the Distributive Law (Theorem 1, page 676). Ex

Question

Must show all your work!!!

Verify the Distributive Law (Theorem 1, page 676). Example 1, page 676, verifies the law for three particular vectors. But now the idea is to verify it for all vectors. So, let u rightarrow = (a1, b1, c1), v rightarrow = (a2, b2,c2)1 and w rightarrow = (a3, b3, c3). Then, just as in Example 1, page 676, first compute u rightarrow . (v rightarrow + w rightarrow), then compute u rightarrow .v rightarrow + u rightarrow . w rightarrow , and then compare the results to make sure they are the same.

Explanation / Answer

Sorry for not including the bars over vectors,

u=(a1,b1,c1) v=(a2,b2,c2) w=(a3,b3,c3)

v+w=(a2+a3 , b2+b3 , c2+c3)

u.(v+w) = (a1,b1,c1).(a2+a3 , b2+b3 , c2+c3) = (a1*(a2+a3) , b1*(b2+b3) , c1*(c2+c3))

=(a1*a2+a1*a3 , b1*b2+b1*b3 , c1*c2+c1*c3) -----------(1)

u.v = (a1,b1,c1).(a2,b2,c2)=(a1*a2 , b1*b2 , c1*c2) -------(2)

u.w = (a2,b2,c2).(a3,b3,c3) = (a2*a3 , b2*b3 , c2*c3) ------(3)

u.v+u.w= (1)+(2) = (a1*a2 , b1*b2 , c1*c2) + (a2*a3 , b2*b3 , c2*c3)

= (a1*a2+a1*a3 , b1*b2+b1*b3 , c1*c2+c1*c3) --------(4)

Clearly we can see that (1)=(4)

So, u.v+u.w = u.(v+w)

Hence we have proved required result.

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