At any time, you may reset the applet by pressing the refresh symbol in the top right. You can zoom out the applet by using your scroll wheel on your mouse inside the applet. You can drag inside the applet to see more to each side as well.[b][br][br]Task 1:[/b][br]You must use the given vectors [math]\vec{x}_1[/math] and [math]\vec{x}_2[/math]. The only things that you can adjust are the weights [math]c_1[/math] and [math]c_2[/math] by moving the sliders left and right.[br][br](a) Can you find weights such that [math]c_1\vec{x}_1+c_2\vec{x}_2[/math] is the vector (8,1)? [br](b) Are your weights from (a) unique, or are there other choices? [br](c) Are you able to get to any point on the plane using the given vectors?[br]If there are other choices, what is another choice?[br](d) Can you find weights such that [img]data:image/png;base64,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[/img] is the zero vector?[br][br][b]Task 2:[/b][br]Now you may choose your own vectors [math]\vec{x}_1[/math] and [math]\vec{x}_2[/math] by sliding the blue and green vertices, respectively. You can also still adjust the weights [math]c_1[/math] and [math]c_2[/math] by moving the sliders left and right.[br][br](a) Can you give an example of a linear combination that resulted in the zero vector that did [u]not[/u] have 0's for the weights.[br](b) What is interesting about your vectors [math]\vec{x}_1[/math] and [math]\vec{x}_2[/math] from (b)?