GATE ECE 2016 Set 3 | Question: 27

Question

If the vectors $e_1=(1,0,2)$, $e_2=(0,1,0)$ and $e_3=(-2,0,1)$ form an orthogonal basis of the three-dimensional real space $\mathbb{R}^3$, then the vector $\textbf{u}=(4,3,-3)\in \mathbb{R}^3 $ can be expressed as

• A. $\textbf{u}=-$$\large\frac{2}{5}$$e_1-3e_2-$$\large\frac{11}{5}$$e_3\\$
• B. $\textbf{u}=-$$\large\frac{2}{5}$$e_1-3e_2+$$\large\frac{11}{5}$$e_3 \\$
• C. $\textbf{u}=-$$\large\frac{2}{5}$$e_1+3e_2+$$\large\frac{11}{5}$$e_3 \\$
• D. $\textbf{u}=-$$\large\frac{2}{5}$$e_1+3e_2-$$\large\frac{11}{5}$$e_3$

Answer 1

$u= e_1* x_1+e_2*x_2+e_3*x_3$

$(4,3,-3)=(1,0,2)* x_1+(0,1,0)*x_2+(-2,0,1)*x_3$

$x_1-2 x_3=4$

$x_2=3$

$2x_1+x_3=-3$

$x_1=\frac{-2}{5}$

$x_3=\frac{-11}{5}$

$u= -\frac{2}{5}e_1+3e_2-\frac{11}{5}e_3$

$Ans= D$