When two column vector in R2 are equal? Give an example. Computer u + 3v, u – 2v, where

u = \(\begin{bmatrix}1\\ -3\\ 2\end{bmatrix}\), v = \(\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}\)

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Solution:

Given,

u = \(\begin{bmatrix}1\\ -3\\ 2\end{bmatrix}\)

v = \(\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}\)

Now,

u + 3v = \(\begin{bmatrix}1\\ -3\\ 2\end{bmatrix}\) + 3 \(\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}\)

= \(\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}\) + \(\begin{bmatrix}3\\ -3\\ 9\end{bmatrix}\)

= \(\begin{bmatrix}1+3\\ -1-3\\ 3+9\end{bmatrix}\)

= \(\begin{bmatrix}4\\ -4\\ 12\end{bmatrix}\)

Again,

u – 2v = \(\begin{bmatrix}1\\ -3\\ 2\end{bmatrix}\) – 2 \(\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}\)

= \(\begin{bmatrix}1\\ -3\\ 2\end{bmatrix}\) – \(\begin{bmatrix}2\\ -2\\ 6\end{bmatrix}\)

= \(\begin{bmatrix}1-2\\ -3-2\\ 2-6\end{bmatrix}\)

= \(\begin{bmatrix}-1\\ -5\\ -4\end{bmatrix}\)

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