If we start with two vectors, [math]\vec{u}[/math] and [math]\vec{v}[/math] that are not parallel to each other, we can write any other vector as a linear combination of [math]\vec{u}[/math] and [math]\vec{v}[/math].[br][br]We can think of our usual coordinate plane as being defined by vectors [math]\vec{i}[/math] and [math]\vec{j}[/math]. If we create a new plane, using [math]\vec{u}[/math] and [math]\vec{v}[/math], it will be easy to see how we can use them to name other vectors.[br][br]First, click "Change Vectors" to change [math]\vec{u}[/math] and [math]\vec{v}[/math] from [math]\vec{i}[/math] and [math]\vec{j}[/math] to two new vectors.[br]Now, click "Change Grid" to create a coordinate system based on these new vectors. Notice that going "Right" on this grid means adding another copy of [math]\vec{u}[/math], while going "Up" adds another copy of [math]\vec{v}[/math][br]Click the box next to "Animate" to change the coordinate system. Going "Left" and "Down" would correspond to subtracting them.[br][br]Write the following in terms of [math]\vec{u}[/math] and [math]\vec{v}[/math]:[br][list=1][*][math]\vec{a}[/math][br][/*][*][math]\vec{b}[/math][br][/*][*][math]\vec{c}[/math][br][/*][*][math]\vec{d}[/math][br][/*][*][math]\vec{e}[/math][br][/*][/list][br]You can check your work by clicking "Show linear combination" and typing in the coefficients for [math]\vec{u}[/math] and [math]\vec{v}[/math] to see if you got it correct.[br][br]6-10: Given that [math]\vec{u}=3\vec{i}+\vec{j}[/math] and [math]\vec{v}=2\vec{i}+5\vec{j}[/math], express each vector above in terms of [math]\vec{i}[/math] and [math]\vec{j}[/math]. You can check your work by clicking "Show Original Grid" to overlay the usual x-y coordinate system onto the diagram.