To introduce the scalar product, we will begin with vector projections.[br][br]If vector [math]v[/math] is the "ground", with the sun directly above, and vector [math]u[/math] is a stick in the ground, a "shadow" will be created from vector [math]u[/math] onto vector [math]v[/math]. This "shadow" is called the [b]projection of [math]u[/math][/b] [b]onto [math]v[/math][/b].[br][br]The applet below shows two vectors, which are placed at the same initial point (since we can freely move vectors). You can move around the points, and then use the slider to create the projection of [math]u[/math] onto [math]v[/math]. You can also push "always show" to make the projection not reset.[br][br]Answer the questions, and move around the points to check various cases, until you are convinced of the property.

[b]Question 1:[/b] If the two vectors are acute, what is the relationship between the projection and the vector [math]v[/math]? What if the two vectors are obtuse?[br][br][b]Question 2:[/b] When is the projection equal to the zero vector?[br][br][b]Question 3: [/b]When is the projection equal to [math]u[/math]?[br][br][b]Question 4:[/b] Let the angle between the two vectors be [math]\theta[/math], and the two vectors be acute. Let the length of vector [math]u[/math] have notation [math]\left|u\right|[/math]. Find the length of the projection. [br][br]Now, we created a projection vector using the two vectors, with the same direction as vector [math]v[/math]. Next, we will define the [color=#ff0000]scalar product[/color] ([math]u\cdot v[/math]) as the product of lengths of vector [math]v[/math] and the length of the projection of [math]u[/math] onto [math]v[/math].[br][br][b]Question 5:[/b] Using this definition and your answer to question 4, write out an equation for [math]u\cdot v[/math].