# If $\overrightarrow a$ is any non-zero vector ,then $(\overrightarrow a.\hat i)\hat i\;+(\overrightarrow a.\hat j)\hat j\;+(\overrightarrow a.\hat k)\hat k$ equals__________.

Toolbox:
• $\overrightarrow a=a_1 \hat i+a_2 \hat j+a_3 \hat k$
• $\hat i.\hat i=\hat j.\hat j=\hat k.\hat k=1$
Let $\overrightarrow a=a_1\hat i + a_2\hat j + a_3\hat k$
We know that $(\overrightarrow a.\hat i) \hat i=[(a_1\hat i + a_2\hat j + a_3\hat k).\hat i]\hat i$
$\qquad\qquad\qquad\quad=a_1 \hat i$
Similarly $(\overrightarrow a.\hat j) \hat j=[(a_1\hat i + a_2\hat j + a_3\hat k).\hat j]\hat j$
$\qquad\qquad\qquad\quad=a_2 \hat j$
Similarly $(\overrightarrow a.\hat k) \hat k=[(a_1\hat i + a_2\hat j + a_3\hat k).\hat k]\hat k$
$\qquad\qquad\qquad\quad=a_3 \hat k$
Therefore $(\overrightarrow a.\hat i)\hat i+(\overrightarrow a.\hat j)\hat j+(\overrightarrow a.\hat k)\hat k= a_1\hat i + a_2\hat j + a_3\hat k$
$= \overrightarrow a$
Hence $(\overrightarrow a.\hat i)\hat i+(\overrightarrow a.\hat j)\hat j+(\overrightarrow a.\hat k)\hat k=\overrightarrow a$