Question

i^(i^ * a vector)+j^(j^ * a vector)+k^(k^ * a vector)=

Answer 1

$\hat{i} (\hat{i} +\vec{A}) +\hat{j} (\hat{j} +\vec{A}) +\hat{k} (\hat{k} +\vec{A})$

$\rule{200}{2}$

Before solving this we need to know some important points :-

They are :-

$\hat{i}. \hat{i}=|\hat{i}||\hat{i}|cos0\\ \\ \\ \hat{i}. \hat{i}= 1×1×1=1$

Similarly :-

$\hat{j}. \hat{j}=1\\ \\ \\ \hat{k}. \hat{k}=1$

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And:-

$\hat{i}.\hat{j}= |\hat{i} ||\hat{j}|cos90\\ \\ \\ \hat{i}. \hat{i}=1×1×0=0$

Similarly :-

$\hat{j}. \hat{k}=0\\ \\ \\ \hat{k}. \hat{i}=0$

$\rule{200}{2}$

Now,

Back to question :-

$\hat{i} (\hat{i} +\vec{A}) +\hat{j} (\hat{j} +\vec{A}) +\hat{k} (\hat{k} +\vec{A})$

$\rule{200}{2}$

We have :-

$\hat{i}. \hat{i}+\vec{A} \hat{i} +\\ \\ \hat{j}. \hat{j}+\vec{A} \hat{j}+\\ \\ \hat{k}. \hat{k}+\vec{A} \hat{k}$

On Simplification :-

$=1+\vec{A}\hat{i}+1+\vec{A}\hat{j}+1+\vec{A}\hat{k}\\ \\ \\=3+\vec{A}\hat{i}+\vec{A}\hat{j}+\vec{A}\hat{k}$

$\rule{200}{2}$