Q. 58

Question

Prove part (b) of Theorem 10.8 for vectors in $R^{3}$ ; that is, show that for $u = (u_{1}, u_{2}, u_{3}), v = (v_{1}, v_{2}, v_{3}) and w = (w_{1}, w_{2}, w_{3})$ , $(u + v) + w = u + (v + w)$ .

Step-by-Step Solution

Verified

Answer

It is proved that $(u + v) + w = u + (v + w)$ .

1Step 1: Consider vectors in R 3

Let us consider,

$u = i + j + k v = i - j - k w = 2 i + 2 j + k$

2Step 2: Find LHS and RHS

$LHS = (u + v) + w = (i + j + k + i - j - k) + (2 i + 2 j + k) = (2 i) + (2 i + 2 j + k) = 4 i + 2 j + k RHS = u + (v + w) = (i + j + k) + (i - j - k + 2 i + 2 j + k) = (i + j + k) + (3 i + j) = 4 i + 2 j + k Here, L H S = R H S$

Hence, proved.

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Q. 59

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