Let [color=#c51414] [i]P[/i][/color] be a point on a plane [color=#555][math]\alpha[/math][/color]. [br]Draw the line [i]r[/i], perpendicular to plane [color=#555][math]\alpha[/math][/color] in [i][color=#c51414]P[/color][/i], then a new line [color=#c51414][i]t[/i][/color] in the same plane.[br]Now draw the line [color=#1551b5][i]s[/i][/color], perpendicular to [color=c51414][i]t[/i][/color] through [i][color=#c51414]P[/color][/i].[br]Draw the plane [color=#1551b5][math]\beta[/math][/color], through [i]r[/i] and [color=#1551b5][i]s[/i][/color].[br][br][math]\Longrightarrow[/math] the plane[color=#1551b5] [math] \beta[/math][/color] is perpendicular to line [color=c51414][i]t[/i][/color].[br][br]Move the points in the construction and explore the theorem.

On the left you can see a right triangle [color=#3c78d8][b][i]ABC[/i] [/b][/color]and the [i]squares [/i]built on its sides. [br]You can change the triangle by dragging its vertex [color=#38761d][b][i]C[/i][/b][/color].[br]Use the slider [color=#e06666][b][i]h[/i][/b][/color] to build prisms having those squares as bases.[br][br]Explore the construction by interacting with the point and the slider.[br][color=#1551b5]What is the relation between the volumes of the prisms?[/color][br]

Given three vectors [color=#ff0000][i]u[/i][/color], [i][color=#008800]v[/color] [/i]and [color=#0000ff][i]w[/i][/color] and three scalars [i]a[/i], [i]b [/i]and [i]c[/i], the [i][color=#1551b5]linear combination[/color][/i] of those vectors is vector [i]d[/i] = [i]a[color=#ff0000]u[/color][/i] + [i]b[color=#008800]v[/color][/i] + [i]c[color=#0000ff]w[/color][/i] [br][br]In this applet [color=#ff0000][i]u[/i]=(255,0,0)[/color], [color=#008800][i]v[/i]=(0,255,0)[/color] and [color=#0000ff][i]w[/i] = (0,0,255)[/color], and the scalars [i]a[/i], [i]b [/i]and [i]c [/i]range in [0,1].[br]Move the sliders and watch the resulting position of the linear combination vector.[br][br]Now let's consider this vectorial problem from a different point of view. [i][color=#ff0000]R[/color][color=#008800]G[/color][color=#0000ff]B[/color] [/i]is an [color=#0000ff][i]additive colour model[/i][/color] used to represent the colours of objects in many environments.[br][br]In computers, a [i][color=#ff0000]R[/color][color=#008800]G[/color][color=#0000ff]B[/color][/i] colour is usually stored as a [color=#0000ff][i]triplet of integer numbers[/i][/color] in the range 0 to 255.[br][color=#ff0000](255,0,0)[/color] is [color=#ff0000][i]red [/i][/color]- [color=#008800](0,255,0)[/color] is [color=#008800][i]green [/i][/color]- [color=#0000ff](0,0,255)[/color] is [i][color=#0000ff]blue[/color][/i]. We can view those colours as the [i]vectors [/i]in 3D space, having those components.[br][br]Any other colour is obtained as a [color=#0000ff][i]linear combination [/i][/color]of these basic colours. If you [i]multiply [/i]a [i]vector [/i]by a [i]scalar[/i], you obtain the [i]same colour[/i], but a [i]different shade[/i] of it.[br][br]Move the sliders and find out where the [i]yellows[/i], or the [i]grey [/i]shades are, and examine the position of the corresponding vector in the visual representation.[br][br]Please note that the components of the resulting vector have been [i]rounded[/i], to match the computer requirements whenever you need to enter [i]RGB [/i]triplets to define a colour.[br]You can also use this applet as a reference to [color=#0000ff][i]define dynamic colours[/i][/color] of objects in GeoGebra.