Question

Show that for any vector \(v\) in \(\mathbb{R}^{3}\),

\(v = (v · i)i + (v · j)j + (v · k)k\).

Short answer

It is shown that for any vector \(v\) in \(\mathbb{R}^{3}\), \(v = (v · i)i + (v · j)j + (v · k)k\).

Step-by-step solution

Step 1. Show

We have to show that \(v = (v · i)i + (v · j)j + (v · k)k\). Now, as we know vector \(v\) is in \(\mathbb{R}^{3}\).

Let \(v=ai+bj+ck\)

So, \(v\cdot i=a\left ( 1 \right )\)

\(v\cdot i=a\) ......(a)

And

\(v\cdot j=b\left ( 1 \right )\)

\(v\cdot j=b\) .......(b)

And

\(v\cdot k=c\left ( 1 \right )\)

\(v\cdot k=c\) ........(c)

Step 2. Show

Now, take R.H.S of \(v = (v · i)i + (v · j)j + (v · k)k\) and by using equations (a), (b), and (c) we get,

\((v\cdot i)i + (v\cdot j)j + (v\cdot k)k=ai+bj+ck\)

\((v\cdot i)i + (v\cdot j)j + (v\cdot k)k=v\)

Hence proved.