In this activity, we will look at linear combinations of a pair of vectors, with weights...

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In this activity, we will look at linear combinations of a pair of vectors, with weights a and b. --8--8 The diagram available at http://gvsu.edu/s/0Je can be used to construct linear combinations whose weights a and b may be varied using the sliders at the top. The vectors v and w are drawn in gray while the linear combination is in red. av+bw a. The weight b is initially set to 0. Explain what happens as you vary a with b = 0? How is this related to scalar multiplication? b. What is the linear combination of v and w when a = 1 and b = -2? You may find this result using the diagram, but you should also verify it by computing the linear combination. c. Describe the vectors that arise when the weight b is set to 1 and a is varied. How is this related to our investigations in the preview activity? d. Can the vector e. Can the vector * [8] b [3] be expressed as a linear combination of v and w? If so, what are weights a and b? be expressed as a linear combination of v and w? If so, what are weights a and b? f. Verify the result from the previous part by algebraically finding the weights a and b that form the linear combination g. Can the vector [13] be expressed as a linear combination of v and w? What about the vector [ 15.2 h. Are there any two-dimensional vectors that cannot be expressed as linear combinations of v and w? In this activity, we will look at linear combinations of a pair of vectors, with weights a and b. --8--8 The diagram available at http://gvsu.edu/s/0Je can be used to construct linear combinations whose weights a and b may be varied using the sliders at the top. The vectors v and w are drawn in gray while the linear combination is in red. av+bw a. The weight b is initially set to 0. Explain what happens as you vary a with b = 0? How is this related to scalar multiplication? b. What is the linear combination of v and w when a = 1 and b = -2? You may find this result using the diagram, but you should also verify it by computing the linear combination. c. Describe the vectors that arise when the weight b is set to 1 and a is varied. How is this related to our investigations in the preview activity? d. Can the vector e. Can the vector * [8] b [3] be expressed as a linear combination of v and w? If so, what are weights a and b? be expressed as a linear combination of v and w? If so, what are weights a and b? f. Verify the result from the previous part by algebraically finding the weights a and b that form the linear combination g. Can the vector [13] be expressed as a linear combination of v and w? What about the vector [ 15.2 h. Are there any two-dimensional vectors that cannot be expressed as linear combinations of v and w?