Chapter 1.1, Problem 1PS

To determine

(a) To Describe:

Geometrically the linear combination of the vectors $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\text{ and }\left[\begin{array}{c}3\\ 6\\ 9\end{array}\right]$.

Answer to Problem 1PS

Geometrically the vectors $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\text{ and }\left[\begin{array}{c}3\\ 6\\ 9\end{array}\right]$ represent a line.

Explanation of Solution

Given:

The vectors $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\text{ and }\left[\begin{array}{c}3\\ 6\\ 9\end{array}\right]$.

Since,

   $\left[\begin{array}{c}3\\ 6\\ 9\end{array}\right]=3\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

That is., the given vector is some scalar times of the other vector.

So, geometrically they represent a line.

To determine

(b) To Describe:

Geometrically the linear combination of the vectors $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\text{ and }\left[\begin{array}{c}0\\ 2\\ 3\end{array}\right]$.

Answer to Problem 1PS

Geometrically the vectors $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\text{ and }\left[\begin{array}{c}0\\ 2\\ 3\end{array}\right]$ represent a plane.

Explanation of Solution

Given:

The vectors $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]\text{ and }\left[\begin{array}{c}0\\ 2\\ 3\end{array}\right]$.

Since, there does not any real number c such that,

   $\left[\begin{array}{c}0\\ 2\\ 3\end{array}\right]=c\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$

That is., the vectors cannot be written as a linear combination of each other.

So, geometrically they represent a plane.

To determine

(c) To Describe:

Geometrically the linear combination of the vectors $\left[\begin{array}{c}2\\ 0\\ 5\end{array}\right]\text{,}\left[\begin{array}{c}0\\ 2\\ 2\end{array}\right]\text{ and }\left[\begin{array}{c}2\\ 2\\ 3\end{array}\right]$.

Answer to Problem 1PS

Geometrically the vectors $\left[\begin{array}{c}2\\ 0\\ 5\end{array}\right]\text{,}\left[\begin{array}{c}0\\ 2\\ 2\end{array}\right]\text{ and }\left[\begin{array}{c}2\\ 2\\ 3\end{array}\right]$ represent a plane.

Explanation of Solution

Given:

The vectors $\left[\begin{array}{c}2\\ 0\\ 5\end{array}\right]\text{,}\left[\begin{array}{c}0\\ 2\\ 2\end{array}\right]\text{ and }\left[\begin{array}{c}2\\ 2\\ 3\end{array}\right]$.

Since, there does not any real number c and d such that,

   $\left[\begin{array}{c}2\\ 2\\ 3\end{array}\right]=c\left[\begin{array}{c}2\\ 0\\ 5\end{array}\right]+d\left[\begin{array}{c}0\\ 2\\ 2\end{array}\right]$

The given vectors cannot be written as a linear combination of each other.

So, geometrically they represent a plane.