WF 3.2.3 & WF 3.2.4

Author:: Albert Navetta

If two vectors

and

span

, then for any vector

, it is possible to find scalars

and

such that

The vector

may correspond to any point in the plane. Geometrically this means that starting from the origin, one can move in the directions of

and

(or

and

) and end up at any given point in the plane. We can explore how this is done by picking a target point in the plane and then using geometric reasoning try to find scalars

and

to take us close to the target point. Since we are only estimating, we may not get there exactly. After doing these geometrical experiments, you may want to think about devising a mathematical method for determining exact values for

and

It is easy to find the scalars

and

that work when the initial vectors are

and

You can use the mouse to change the vectors by dragging their tips to different positions. Alternatively, you can create new vectors by changing the coordinates that are displayed. You can also change the coordinates of the target point. Starting from the origin, if we move in the directions of

and

, can we end up at the target point? Can we find scalars

and

such that the tip of the vector

will lie in the circle representing the target point? Try to guess approximate values of

and

and enter your guesses in the text boxes below the graph on the right. To see how well they work, click on the Draw Vector button. If the tip of the resulting vector lies in the target circle, we will consider the approximation to be good enough. If the tip does not lie in the target circle, enter new values for

WF 3.2.3 & WF 3.2.4

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