Linear Combination of Vectors

If we start with two vectors, and that are not parallel to each other, we can write any other vector as a linear combination of and . We can think of our usual coordinate plane as being defined by vectors and . If we create a new plane, using and , it will be easy to see how we can use them to name other vectors. First, click "Change Vectors" to change and from and to two new vectors. 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 , while going "Up" adds another copy of Click the box next to "Animate" to change the coordinate system. Going "Left" and "Down" would correspond to subtracting them. Write the following in terms of and :
You can check your work by clicking "Show linear combination" and typing in the coefficients for and to see if you got it correct. 6-10: Given that and , express each vector above in terms of and . You can check your work by clicking "Show Original Grid" to overlay the usual x-y coordinate system onto the diagram.