- WELCOME TO THE FIRST OF TWO VIDEOS
ON PARALLEL AND PERPENDICULAR LINES.
THE GOALS OF THIS VIDEO ARE TO DEFINE
PARALLEL AND PERPENDICULAR LINES AND ALSO TO DETERMINE
IF TWO LINES ARE PARALLEL OR PERPENDICULAR
FROM THEIR EQUATIONS.
PARALLEL LINES ARE TWO LINES THAT NEVER INTERSECT.
SO LET'S START BY TAKING A LOOK AT TWO LINES GRAPHED TOGETHER
ON THE COORDINATE PLANE.
IF WE TAKE LOOK AT THIS RED LINE AND BLUE LINE
THEY ARE OBVIOUSLY NOT PARALLEL BECAUSE THEY INTERSECT.
SO WHAT WE'RE GOING TO DO HERE
IS CHANGE THE SLOPE OF THE BLUE LINE
UNTIL IT'S PARALLEL WITH THE RED LINE
AND BEFORE WE DO THAT NOTICE THAT THE SLOPE OF THE RED LINE
IS POSITIVE BECAUSE IT MOVES UP FROM LEFT TO RIGHT.
SO IN ORDER FOR THE BLUE LINE TO BE PARALLEL
IT MUST ALSO BE SLANTED IN THE SAME DIRECTION
SO IT MUST ALSO HAVE A POSITIVE SLOPE.
SO AS WE INCREASE THE SLOPE OF THE BLUE LINE,
THE SLOPE OF THE BLUE LINE WILL APPROACH
THE SLOPE OF THE RED LINE.
SO RIGHT HERE, THE SLOPE OF THE RED LINE = 2
AND THE SLOPE OF THE BLUE LINE = 2.
SO IF THE SLOPES ARE THE SAME THE TWO LINES WILL BE PARALLEL
EXCEPT FOR ONE CASE.
NOTICE IF WE CHANGE THE Y INTERCEPT OF THE BLUE LINE,
IT CONTINUES TO BE PARALLEL TO THE RED LINE EXCEPT IN ONE CASE.
IF THE Y INTERCEPTS HAPPEN TO BE THE SAME
AND THE SLOPES ARE THE SAME,
OF COURSE THAT WOULD BE THE SAME LINE,
WHICH HAS AN INFINITE NUMBER OF INTERSECTION POINTS.
SO THIS LEADS US TO THE CONDITIONS
THAT MUST BE MET IN ORDER TO HAVE PARALLEL LINES.
THE SLOPES MUST EQUAL EACH OTHER
BUT THE Y INTERCEPTS MUST BE DIFFERENT.
SO AGAIN, TWO LINES ARE PARALLEL IF THEY HAVE THE SAME SLOPE
AND DIFFERENT Y INTERCEPTS.
ANOTHER WAY TO INDICATE THIS IF THE LINES ARE IN SLOPE INTERCEPT
FORM IS THAT M1,
THE SLOPE OF LINE 1 MUST EQUAL THE SLOPE OF LINE 2
BUT THE TWO Y INTERCEPTS B SUB 1 AND B SUB 2 MUST BE DIFFERENT.
LET'S TAKE A LOOK AT THIS ONE MORE TIME.
HERE WE HAVE TWO DIFFERENT LINES,
LET'S JUST DETERMINE THE SLOPE OF EACH OF THEM.
LOOKING AT THE BLUE LINE, IF WE PICK TWO CONVENIENT POINTS,
LET'S SAY THIS POINT HERE AND THIS POINT HERE,
WE CAN SEE THE SLOPE WOULD BE UP 1 RIGHT 2.
SO THE SLOPE OF THE BLUE LINE IS EQUAL TO 1/2
AND THE SLOPE OF THE RED LINE IF WE GO UP 1 AND RIGHT 2,
IS ALSO EQUAL TO 1/2.
AND NOTICE THAT THE Y INTERCEPT OF THE BLUE LINE IS EQUAL TO 3.
AND THE Y INTERCEPT OF THE RED LINE IS EQUAL TO -2.
SO THE SLOPES ARE THE SAME.
THE Y INTERCEPTS ARE DIFFERENT SO WE HAVE PARALLEL LINES.
NOW LET'S TALK ABOUT PERPENDICULAR LINES.
PERPENDICULAR LINES ARE TWO LINES
THAT INTERSECT AND FORM A RIGHT ANGLE.
LET'S GO BACK TO OUR 2 LINES GRAPHED ON THE COORDINATE PLANE.
AGAIN, WE'RE GOING TO CHANGE THE SLOPE OF THE BLUE LINE
UNTIL IT APPEARS THAT IT WILL PERPENDICULAR TO THE RED LINE.
REMEMBER THAT MEANS THAT THEY INTERSECT
AND FORM A RIGHT ANGLE.
SO THEY'RE GETTING CLOSER AND CLOSER TO BEING PERPENDICULAR.
I'M NOT SURE IF YOU CAN SEE IT,
BUT THE SLOPE OF THE RED LINE = 2
AND THE SLOPE OF THE BLUE LINE = -0.5.
AND THESE TWO LINES ARE PERPENDICULAR.
ANOTHER WAY TO SAY THAT IS THAT
THE SLOPE OF THE RED LINE WOULD BE 2/1
AND THE SLOPE OF THE BLUE LINE WOULD BE -1/2 OR -1/2,
AND THAT LEADS US TO THE CONDITION THAT MUST BE MET
FOR TWO LINES TO BE PERPENDICULAR.
THEIR SLOPES MUST BE NEGATIVE RECIPROCALS OF ONE ANOTHER
AND IF WE MULTIPLY THE TWO TOGETHER, IT MUST EQUAL -1.
LET'S GO AHEAD AND STATE THAT AGAIN.
TWO LINES ARE PERPENDICULAR IF THE PRODUCT OF THE SLOPES IS -1
OR IF THE SLOPES ARE NEGATIVE RECIPROCALS OF ONE ANOTHER.
SO IF BOTH LINES ARE IN SLOPE INTERCEPT FORM,
M1 x M2 MUST EQUAL -1 OR M1 MUST EQUAL -1/M2.
NOW THESE CONDITIONS MAY SEEM A LITTLE MORE INVOLVED
THAN PARALLEL LINES BUT LOOKING AT A NUMERICAL EXAMPLE
IS OFTEN HELPFUL.
SO FOR EXAMPLE, IF THE SLOPE OF LINE 1 WAS EQUAL TO 2/3
AND IT WAS PARALLEL TO LINE 2
THEN M2 MUST BE THE NEGATIVE RECIPROCAL.
SO WE FLIP IT OVER AND CHANGE THE SIGN.
AND THIS WOULD GUARANTEE THAT THOSE TWO LINES
ARE PERPENDICULAR.
NOTICE THAT IF WE DID FIND THE PRODUCT OF THESE TWO SLOPES
WE WOULD HAVE -6/6 WHICH IS EQUAL TO -1 AS IT STATES HERE.
AND LET'S GO AND TAKE A LOOK AT ONE MORE GRAPH.
FIRST WE SHOULD RECOGNIZE THAT THESE TWO LINES
ARE PERPENDICULAR,
MEAN THEY FORM A RIGHT ANGLE AT THEIR INTERSECTION.
LET'S GO AHEAD AND FIND THE SLOPES OF BOTH LINES.
LOOKING AT THE BLUE LINE HERE,
LET'S GO AHEAD AND USE THIS POINT HERE
AND THIS POINT HERE TO DETERMINE THE SLOPE.
SO IT LOOKS LIKE WE'RE GOING TO GO UP 2, AND RIGHT 4,
SO THAT MEANS THE SLOPE WOULD BE 2/4 OR 1/2.
LET'S DO THE SAME FOR THE RED LINE.
LET'S GO AHEAD AND USE THESE TWO POINTS HERE.
WELL NOW WE'LL GO DOWN 2, AND RIGHT 1.
SO THE SLOPE OF THE RED LINE IS EQUAL TO -2/1.
NOW NOTICE IN THIS FORM IT'S EASY TO SEE
THAT THESE TWO ARE NEGATIVE RECIPROCALS OF ONE ANOTHER.
MEANING IF WE TAKE THIS SLOPE HERE, FLIP IT OVER,
AND CHANGE THE SIGN IT'S THE SLOPE OF THE RED LINE
AND THEREFORE, THESE TWO LINES ARE PERPENDICULAR.
LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE PROBLEMS
WHERE WE'RE GIVEN THE EQUATION OF TWO LINES
AND WE'RE ASKED TO DETERMINE IF THE LINES ARE PARALLEL,
PERPENDICULAR, OR NEITHER.
SO THE FIRST THING WE SHOULD NOTICE IS THAT
IN THIS FIRST LINE THE SLOPE IS EQUAL TO 2/3
AND THE Y INTERCEPT IS EQUAL TO -4.
LET'S GO AHEAD AND SOLVE THE SECOND EQUATION FOR Y
SO WE CAN START BY COMPARING THE SLOPES.
SO IF WE WANT TO SOLVE THIS EQUATION FOR Y
WE SHOULD ADD 2X TO BOTH SIDES.
SO WE HAVE 3Y = 2X + 3, NOW WE'LL DIVIDE EVERYTHING BY 3.
SO WE HAVE Y = 2/3X + 1.
WELL THE SLOPE OF THIS LINE IS EQUAL TO 2/3 AS WELL
AND THE Y INTERCEPT IS EQUAL TO 1.
SO NOTICE THAT THE SLOPES ARE THE SAME,
WHICH MEANS THEY WILL BE PARALLEL
AS LONG AS THE Y INTERCEPTS ARE DIFFERENT
AND THE Y INTERCEPTS ARE DIFFERENT.
THIS Y INTERCEPT IS -4 AND THIS Y INTERCEPT IS 1.
SO THE LINES ARE PARALLEL.
LET'S GO AHEAD AND VERIFY THIS GRAPHICALLY AS WELL
BY GRAPHING THIS FIRST LINE
AND THE SECOND LINE ON THE GRAPHING CALCULATOR
AT THE SAME TIME.
SO WE'LL PRESS Y = AND WE'RE GOING TO TYPE IN 2/3X - 4.
SO WE'RE GOING TO PUT THE FRACTIONAL SLOPE IN PARENTHESES.
x X - 4 THAT'S OUR FIRST LINE, ENTER.
AND THEN FOR OUR SECOND LINE
WE'LL DO THE SAME THING (2/3X + 1).
NOW TO MAKE SURE WE ALL HAVE THE STANDARD WINDOW
LET'S GO AHEAD AND PRESS ZOOM 6.
THERE'S THE FIRST LINE, THERE'S THE SECOND LINE
AND WE CAN SEE VERY CLEARLY THESE TWO LINES ARE PARALLEL.
LET'S GO AHEAD AND TAKE A LOOK AT ANOTHER ONE.
SAME QUESTION, NOW NOTICE THAT BOTH OF THESE LINEAR EQUATIONS
ARE IN STANDARD FORM
SO WE WILL HAVE TO SOLVE BOTH EQUATIONS FOR Y.
SO TO SOLVE THIS EQUATION FOR Y WE'LL START BY SUBTRACTING 2X
ON BOTH SIDES.
WE HAVE 5Y = -2X + 9 AND THEN WE'LL DIVIDE EVERY TERM BY 5.
FOR THIS LINE WE HAVE Y = -2/5X + 9/5.
SO THE SLOPE IS -2/5 AND THE Y INTERCEPT IS 9/5.
NOW LET'S GO AHEAD AND SOLVE THE SECOND EQUATION FOR Y.
SO ON THIS ONE WE'LL SUBTRACT 10X ON BOTH SIDES.
DIVIDE EVERYTHING BY -4 AND WE HAVE Y = 5 1/2X.
NOTICE THIS SIMPLIFIES AND THIS WOULD BE -3.
SO WE HAVE A SLOPE OF 5/2 AND WE HAVE A Y INTERCEPT OF -3
AND NOTICE THAT THE SLOPES ARE NEGATIVE RECIPROCALS
THEREFORE THESE LINES ARE PERPENDICULAR TO ONE ANOTHER.
AND AGAIN, LET'S GO AHEAD AND VERIFY THIS GRAPHICALLY.
SO ENTER IN THIS EQUATION FOR Y1 AND THIS EQUATION FOR Y2,
AND SEE IF THEY INTERSECT AND FORM A RIGHT ANGLE.
LET'S CLEAR OUT THE OLD LINES.
THE FIRST LINE WAS Y = -2/5X + 9/5 AND Y2 WAS Y = 5/2X - 3.
NOW IF YOU TAKE A LOOK AT THESE TWO LINES
IT LOOKS LIKE THEY DO NOT INTERSECT IN A RIGHT ANGLE
AND THAT'S BECAUSE THIS SCREEN IS WIDER THAN IT IS TALL.
SO IT'S DISTORTED.
SO WHAT WE HAVE TO DO IS SQUARE THE WINDOW AND WE CAN DO THAT
BY PRESSING ZOOM AND THEN OPTION 5 FOR Z SQUARE.
AND WE CAN NOW SEE THAT THESE TWO LINES DO INTERSECT
IN A RIGHT ANGLE.
OKAY, THAT'S GOING TO DO IT FOR THIS VIDEO.
IN PART TWO, WE'LL TAKE A LOOK AT DETERMINING
THE EQUATIONS OF PARALLEL AND PERPENDICULAR LINES
BASED UPON GIVEN INFORMATION.
I HOPE YOU FOUND THIS VIDEO HELPFUL.