Question

if A is any vector then prove that A=(A.i)i+(A.j)j+(A.k)k

Answer 1

Let \[A = A_x i + A_y j + A_z k\]

Ax, Ay and Az are components of the vector

SO \[A.i = (A_x i + A_y j + A_z k) . i = A_x\]

\[A .j = (A_x i + A_y j + A_z k) . j = A_y\]

\[A.k = (A_x i + A_y j + A_z k) . k = A_z\]

Therfore \[(A.i)i + (A.j) j + (A. k) k = A_x i + A_y j + A_z k=A\]