TIFR ECE 2022 | Question: 10

Question

Find the vector which is closest (in Euclidean distance) to $\left(\begin{array}{lll}-1 & 1 & 1\end{array}\right)$ which can be written in the form

\[a\left(\begin{array}{lll}
1 & 1 & 1
\end{array}\right)+b\left(\begin{array}{lll}
0 & 1 & -1
\end{array}\right)\]

where $a$ and $b$ are some real numbers. Recall that the (Euclidean) distance between two vectors $\left(\begin{array}{llll}x_{1} & x_{2} & x_{3}\end{array}\right)$ and $\left(\begin{array}{lll}y_{1} & y_{2} & y_{3}\end{array}\right)$ is given by $\sum_{i=1}^{3}\left(x_{i}-y_{i}\right)^{2}$.

• A. $\frac{1}{3}\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)$
• B. $\frac{1}{2}\left(\begin{array}{lll}0 & 1 & -1\end{array}\right)$
• C. $\frac{1}{3}\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)+\frac{1}{2}\left(\begin{array}{lll}0 & 1 & -1\end{array}\right)$
• D. $-\frac{1}{3}\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)+\frac{1}{2}\left(\begin{array}{lll}0 & 1 & -1\end{array}\right)$
• E. None of the above