ACTIVITY #6 SLOPE IN TWO AND THREE DIMENSIONS AND VERTICAL CHANGE ON A PLANE

Transcription

1 ACTIVITY # SLOPE IN TWO AND THREE DIMENSIONS AND VERTICAL CHANGE ON A PLANE Name: VERTICAL CHANGE AND SLOPE IN TWO DIMENSIONS The first three problems eplain how to compute slope looking at a figure, WITHOUT USING FORMULAS; visuall one ma determine the sign (increasing-positive, decreasing-negative) of the slope and the magnitudes of vertical and horiontal changes ma be determined b counting In the following figures, each square has the same vertical and horiontal measure of one unit In each case: Determine the sign of the slope of the line Choose two points on the line and find the magnitudes of the vertical and horiontal change from one point to the other (WITHOUT USING FORMULAS) Find the slope of the line (WITHOUT USING FORMULAS) WITHOUT USING FORMULAS, find the slope of each one of the following lines (each square measures one unit verticall and horiontall) WITHOUT USING FORMULAS, find the slope of each one of the following lines (each square measures one unit verticall and horiontall)

2 Since the slope of a line is vertical change over horiontal change, that is, m V / H, then V m H Equivalentl: Vertical change is the slope times the horiontal change In each one of the following lines use the fact that vertical change is slope times horiontal change to compute the vertical change that corresponds to each given horiontal change Line Horiontal change Vertical change H V H V H V H V H V H V SLOPES IN THE X AND Y DIRECTIONS OF A PLANE H V H V H V H V A non-vertical line in space is said to be in the if it is in a plane of the form for some constant c On the plane that is below on the left, darken three lines that are in the c A non-vertical line in space is said to be in the if it is in a plane of the form c for some constant c On the plane that is above to the right, darken three lines that are in the

3 The slope of a line in the is computed as usual, keeping in mind that in three dimensions vertical means up or down, that is, in the : Decide the sign: if when increases increases then the sign is positive; if when increases decreases then the sign is negative Take two points on the line (in the ) The magnitude of the slope is equal to the vertical change d from one point to the other ( final initial ) over the horiontal change d, ( final initial ), from one point to the other The slope of a line in the is defined in a similar manner; vertical change over horiontal change, d / d (where d final initial and d final initial ), with the sign depending on whether increases (positive) or decreases (negative) when increases In each of the following planes identif a line in the and find its slope (use a line with two points with coordinates that ma be determined easil) In each of the following planes identif a line in the and find its slope (use a line with two points with coordinates that ma be determined easil) 9 Let P be the plane with equation a The intersection of P with the fundamental plane is a line with slope m? b The intersection of P with the fundamental plane is a line with slope m? c The intersection of P with the fundamental plane is a line with slope m? d The intersection of P with the fundamental plane is a line with slope m? e The intersection of P with the fundamental plane is a line with slope m? f The intersection of P with the fundamental plane is a line with slope m? It ma be shown that in a plane, all lines in the are parallel and hence have the same slope; this is called m, the slope in the of the plane Similarl, in a plane all lines in the have the same slope which is called plane m, the slope in the of the Find the slope in the and the slope in the for each one of the following planes a b c

4 VERTICAL CHANGE ON A PLANE In the following problem remember that on a line: Vertical change = Slope Horiontal change a Use the formula above to fill the following table, line b line, for the plane given below and the initial point as given in the table b Reflect on what ou did in the previous part and eplain in our own words how to find vertical change on a plane Initial point Horiontal change in d Vertical change in d Horiontal change in d (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) ( abc,, ) d d Vertical change in d Total vertical change d d d

5 The following is a table of values of a plane: a Find m b Find m 9 c Epress the vertical change d as a function of the horiontal change in the, d, and the horiontal change in the, d The following is the contour diagram of a plane: a Find m b Find m c Epress the vertical change d as a function of the horiontal change in the, d, and the horiontal change in the, d