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Question: If **a** is any vector in space, then...

If a is any vector in space, then

A

a=(a.i)i+(a.j)j+(a.k)k\mathbf{a} = (\mathbf{a}.\mathbf{i})\mathbf{i} + (\mathbf{a}.\mathbf{j})\mathbf{j} + (\mathbf{a}.\mathbf{k})\mathbf{k}

B

a=(a×i)+(a×j)+(a×k)\mathbf{a} = (\mathbf{a} \times \mathbf{i}) + (\mathbf{a} \times \mathbf{j}) + (\mathbf{a} \times \mathbf{k})

C

a=j(a.i)+k(a.j)+i(a.k)\mathbf{a} = \mathbf{j}(\mathbf{a}.\mathbf{i}) + \mathbf{k}(\mathbf{a}.\mathbf{j}) + \mathbf{i}(\mathbf{a}.\mathbf{k})

D

a=(a×i)×i+(a×j)×j+(a×k)×k\mathbf{a} = (\mathbf{a} \times \mathbf{i}) \times \mathbf{i} + (\mathbf{a} \times \mathbf{j}) \times \mathbf{j} + (\mathbf{a} \times \mathbf{k}) \times \mathbf{k}

Answer

a=(a.i)i+(a.j)j+(a.k)k\mathbf{a} = (\mathbf{a}.\mathbf{i})\mathbf{i} + (\mathbf{a}.\mathbf{j})\mathbf{j} + (\mathbf{a}.\mathbf{k})\mathbf{k}

Explanation

Solution

Leta=a1i+a2j+a3k,\mathbf{a} = a_{1}\mathbf{i} + a_{2}\mathbf{j} + a_{3}\mathbf{k}, then a.i=a1,\mathbf{a}.\mathbf{i} = a_{1}, a.j=a2,\mathbf{a}.\mathbf{j} = a_{2}, a.k=a3\mathbf{a}.\mathbf{k} = a_{3}

∴a=(a.i)i+(a.j)j+(a.k)k\therefore\mathbf{a} = (\mathbf{a}.\mathbf{i})\mathbf{i} + (\mathbf{a}.\mathbf{j})\mathbf{j} + (\mathbf{a}.\mathbf{k})\mathbf{k}.