[b]Display Starting Vectors[/b][br]Manipulate the sliders and/or input boxes to specify the horizontal and vertical components of vectors [i][b]v[/b][/i] and [i][b]w[/b][/i]. Use the checkbox to hide vector [i][b]w[/b][/i] if it is not desired. Use the checkboxes to show/hide the magnitude and direction angles for these starting vectors and the horizontal and vertical vector components of the two vectors. Use the action buttons to reposition the vectors in standard position with the initial points at the origin.[br][br][b]Vector Sum[/b][br]Use the checkbox to display the vector sum [i][b]v [/b][/i]+ [b][i]w[/i][/b] (in red). Algebraically the components of sum are the sum of the corresponding components of the two original vectors. Graphically addition of vectors is composition of transpositions. Start at any point A and move along vector [i][b]v[/b][/i] to the terminal point. Make this terminal point for [i][b]v[/b][/i] be the initial point for [b][i]w[/i][/b] by using the action button "tip to tail". Slide along vector [b][i]w[/i][/b]. The vector taking the point A to this final terminal point is the vector sum [i][b]v [/b][/i]+ [b][i]w[/i][/b]. [br][br][b]Scalar Multiple[/b][br]Use the checkboxes to hide vector [b][i]w[/i][/b] and show alpha times [i][b]v[/b][/i]. Multiplying the scalar alpha by the vector [i][b]v[/b][/i] produces a vector. Algebraically, we compute this by multiplying alpha by each component of [i][b]v[/b][/i] to get the components of the scalar multiple. Graphically, the magnitude of [math]\alpha[/math][i][b]v [/b][/i]is [math]\left|\alpha\right|\parallel v\parallel[/math], i.e. the length of the original vector is multiplied by the absolute value of the scalar factor. The direction of the scalar multiple is the same as the direction of the original vector if [math]\alpha[/math] is positive, and the direction is the opposite of the direction of the original vector if [math]\alpha[/math] is negative. If [math]\alpha=0[/math], then the product is the zero vector (all components 0), which has no direction.[br]We can similarly show beta times [i][b]w[/b][/i].[br][b][br]Linear Combination[/b][br]We can also use the app to illustrate the linear combination: [math]\alpha[/math][i][b]v [/b][/i]+ [math]\beta[/math][i][b]w[/b][/i] .[br]