Linear Combination of Vectors

Author:: Brian Sterr

Topic:: 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.

Linear Combination of Vectors

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