Below you will see a visual demonstration of the projection of vector [math]\vec{u}[/math] onto the (nonzero) vector [math]\vec{v}[/math]. This is denoted as [math]proj_{\vec{v}}\vec{u}[/math] and is itself a vector. (In the case of this applet, notice that [math]\vec{v}[/math] lies along the x axis.)[br][br]You can think of [math]proj_{\vec{v}}\vec{u}[/math] as the shadow cast by [math]\vec{u}[/math] onto [math]\vec{v}[/math]. The goal is to determine how closely aligned these vectors are. In other words, how much of [math]\vec{u}[/math] points in the same direction as [math]\vec{v}[/math]?[br][br]Please explore this further by moving the [b]♦ [/b]shapes at the end of each vector. This will automatically change the "shadow" of u onto v. Some questions to consider: [br][br]1. For what angle, [math]\theta[/math], does [math]proj_{\vec{v}}\vec{u}[/math] point in the same direction as [math]\vec{v}[/math]?[br]2. For what angle, [math]\theta[/math], does [math]proj_{\vec{v}}\vec{u}[/math], point in the opposite direction as [math]\vec{v}[/math]?[br]3. For what angle, [math]\theta[/math], is there no shadow at all? What does [math]proj_{\vec{v}}\vec{u}[/math] equal in this special case?[br]