I've had a few of these problems come up, and I really don''t understand how to relate the solutions of the problems to the vectors. Maybe someone could give me a good direction to go with this. Help would be much appreciated!
Consider the vectors \(\displaystyle \vec{v_{1}}\), \(\displaystyle \vec{v_{2}}\), \(\displaystyle \vec{v_{3}}\), in \(\displaystyle \mathbb{R}^2\), Vectors \(\displaystyle \vec{v_{1}}\) and \(\displaystyle \vec{v_{2}}\) are parallel. How many solutions x, y, does the system below have? Argue, geometrically.
\(\displaystyle x\vec{v_{1}}+y\vec{v_{2}} = \vec{v_{3}}\)
Consider the vectors \(\displaystyle \vec{v_{1}}\), \(\displaystyle \vec{v_{2}}\), \(\displaystyle \vec{v_{3}}\), in \(\displaystyle \mathbb{R}^2\), Vectors \(\displaystyle \vec{v_{1}}\) and \(\displaystyle \vec{v_{2}}\) are parallel. How many solutions x, y, does the system below have? Argue, geometrically.
\(\displaystyle x\vec{v_{1}}+y\vec{v_{2}} = \vec{v_{3}}\)
