Q. 57

Question

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

Step-by-Step Solution

Verified

Answer

It is proven that $u + v = v + u$

1Step 1: Given

Vectors $u and v$ in $R^{3}$ .

2Step 2: Proof

Let us take two vectors,

$u = 1 i + 2 j + 3 k and v = 4 i + 5 j + 6 k L H S = u + v = (1 i + 2 j + 3 k) + (4 i + 5 j + 6 k) = (1 i + 4 i) + (2 j + 5 j) + (3 k + 6 k) = 5 i + 7 j + 9 k R H S = v + u = (4 i + 5 j + 6 k) + (1 i + 2 j + 3 k) = (1 i + 4 i) + (2 j + 5 j) + (3 k + 6 k) = 5 i + 7 j + 9 k$

It is observed that, $L H S = R H S$

Hence, proved.

Q. 52

Q.10.2

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