WELCOME TO AN INTRODUCTORY VIDEO ON VECTORS.
THE GOAL IS TO DEFINE A VECTOR
AND ALSO TO DEFINE BASIC VECTOR VOCABULARY.
SO MEASUREMENTS THAT INVOLVE
TIME, LENGTH, AREA, VOLUME AND TEMPERATURE
ARE CALLED SCALAR MEASUREMENTS OR SCALAR QUANTITIES
BECAUSE THEY COULD BE ADEQUATELY DESCRIBED
BY THEIR MAGNITUDE ALONE WITH THE APPROPRIATE UNITS.
AND THE RELATED NUMBER IS CALLED A SCALAR.
SO FOR EXAMPLE WHEN WE TALK ABOUT THE TEMPERATURE OUTSIDE
IF IT'S 85 DEGREES,
THAT'S ENOUGH INFORMATION TO DESCRIBE HOW WARM IT IS OUTSIDE.
OTHER MEASUREMENTS THAT REQUIRE MORE
THAN A SINGLE QUANTITY TO DESCRIBE THEIR ATTRIBUTES
ARE CALLED VECTOR QUANTITIES.
EXAMPLES OF VECTOR QUANTITIES ARE FORCE AND VELOCITY.
THESE REQUIRE BOTH MAGNITUDE AND DIRECTION
TO COMPLETELY DESCRIBE.
SO FOR EXAMPLE IF YOU'RE DRIVING YOUR CAR
AND YOU'RE GOING 60 MILES PER HOUR,
YOU WOULD ALSO WANT TO KNOW WHAT DIRECTION YOU'RE DRIVING IN
SO IT TAKES TWO QUANTITIES, MAGNITUDE AND THE DIRECTION.
A DIRECTED LINE SEGMENT IS USED TO REPRESENT A VECTOR QUANTITY
AS WE SEE HERE IN BLACK.
THE LENGTH OF THE VECTOR MODELS THE MAGNITUDE.
THE ARROWHEAD MODELS THE DIRECTION.
THE ORIGIN OF THE SEGMENT IS CALLED THE INITIAL POINT
AND THE ARROWHEAD POINTS TO THE TERMINAL POINT.
SO HERE WE HAVE A VELOCITY VECTOR WITH INITIAL POINT "A,"
TERMINAL POINT B, ITS MAGNITUDE IS 55,
AND THE DIRECTION WOULD BE HEADING EAST.
AND HERE'S A FOURTH VECTOR THAT'S 120 NEWTONS
AND YOU CAN SEE THE FORCE IS SLIGHTLY UPWARD.
YOU CAN ALSO LABEL A VECTOR WITH A BOLD, LOWER CASE LETTER.
IF YOU CAN'T BOLD THE LETTER
YOU SHOULD PUT A LITTLE VECTOR SYMBOL ABOVE THE LETTER
WHICH WE'LL SEE SHORTLY.
VECTORS ARE EQUAL TO EACH OTHER
IF THEY HAVE THE SAME MAGNITUDE AND THE SAME DIRECTION.
SO THESE FIRST TWO VECTORS
EVEN THOUGH THEY HAVE THE SAME MAGNITUDE,
THEY'RE NOT POINTING IN THE SAME DIRECTION THEREFORE,
THEY'RE NOT EQUAL.
THE SECOND PAIR ARE POINTING IN THE SAME DIRECTION
BUT THEY HAVE DIFFERENT MAGNITUDES
BECAUSE THEY HAVE DIFFERENT LENGTHS AND SO THEY'RE NOT EQUAL
BUT THESE LAST PAIR ARE EQUAL BECAUSE THEY HAVE
THE SAME MAGNITUDE AND THE SAME DIRECTION.
A VECTOR WITH ITS INITIAL POINT AT THE ORIGIN
AS WE SEE HERE IN RED, IS CALLED A POSITION VECTOR.
A POSITION VECTOR U WITH ITS END POINT AT THE POINT [A,B]
IS WRITTEN USING THIS NOTATION
AND NOTICE THEY USE THIS TYPE OF BRACKET
INSTEAD OF ROUNDED PARENTHESES
SO IT'S NOT CONFUSED WITH AN ORDERED PAIR
AND THIS IS CALLED COMPONENT FORM.
THE NUMBERS "A" AND B REPRESENT
THE HORIZONTAL AND VERTICAL COMPONENTS OF VECTOR U.
SO NOTICE IN BLACK THIS WOULD BE THE HORIZONTAL COMPONENT.
SO WE CAN LABEL THIS "A"
AND THE VERTICAL COMPONENT IS HERE SO WE CAN LABEL THIS B.
THE POSITIVE ANGLE BETWEEN THE X AXIS AND A POSITION VECTOR
IS THE DIRECTION ANGLE FOR THE VECTOR.
THAT WOULD BE THIS ANGLE HERE.
LET'S GO AHEAD AND TAKE A LOOK AT THIS ANOTHER WAY.
HERE WE SEE A VECTOR IN RED
WITH ITS HORIZONTAL AND VERTICAL COMPONENTS IN BLACK.
AS WE CHANGE THE VECTOR,
THE HORIZONTAL AND VERTICAL COMPONENTS ALSO CHANGE.
AGAIN THIS WOULD BE THETA HERE
AND THE LENGTH OF THIS RED VECTOR
WOULD BE ITS MAGNITUDE.
FOR A VECTOR V WITH AN INITIAL POINT [X1,Y1]
AND A TERMINAL POINT, [X2,Y2]
THE UNIQUE POSITION VECTOR IS VECTOR V = X2 - X1, Y2 - Y1.
THIS IS AN EQUIVALENT VECTOR WITH THE INITIAL POINT 00
AND TERMINAL POINT AS WE SEE HERE.
LET'S GO AHEAD AND WRITE THIS VECTOR AS A POSITION VECTOR.
SO THIS IS THE POINT [-2,-3] THIS IS THE POINT [-4,1].
SO IF THIS IS VECTOR V,
WE CAN WRITE THIS AS A POSITION VECTOR
BY SUBTRACTING THE HORIZONTAL COMPONENTS
SO -4 - -2 AND 1 - -3.
SO THIS WOULD BE [-2,4].
SO IF WE GRAPH THIS POSITION VECTOR
ITS INITIAL POINT WILL BE AT THE ORIGIN
AND IT'S TERMINAL POINT WOULD BE AT [-2,4].
THESE TWO VECTORS ARE EQUAL
BUT THE BLACK VECTOR IS THE POSITION VECTOR.
NEXT LET'S TALK ABOUT THE MAGNITUDE
AND DIRECTION ANGLE OF A VECTOR.
THE MAGNITUDE OR LENGTH OF A VECTOR U IS GIVEN
BY THROUGH THE SQUARE ROOT OF "A" SQUARED + B SQUARED.
AND THE DIRECTION ANGLE THETA,
SATISFIES THE EQUATION TANGENT THETA = B DIVIDED BY "A"
WHERE "A" DOESN'T EQUAL 0. LET'S TAKE A LOOK AT A PROBLEM.
IF WE WANT TO DETERMINE THE MAGNITUDE
AND DIRECTION OF THIS VECTOR,
WE FIRST HAVE TO HAVE IT AS A POSITION VECTOR AND WE DO.
SO NOW WE CAN GO AHEAD AND JUST USE THESE FORMULAS.
MAGNITUDE OF U = TO THE SQUARE ROOT OF 3 SQUARE + 4 SQUARED,
SO IT WOULD BE 9 PLUS 16 OR 25, SO THE MAGNITUDE IS EQUAL TO 5.
BEFORE WE FIND ANGLE THETA,
WE SHOULD MAKE A QUICK SKETCH OF THIS VECTOR
SO WE KNOW WHICH QUADRANT IT IS IN
BECAUSE THAT MAY AFFECT
THE ANGLE WE FIND FROM THIS EQUATION.
THIS WILL BE OUR DIRECTION ANGLE.
SO TANGENT THETA MUST EQUAL B DIVIDED BY "A" OR 4/3.
LET'S GO AHEAD AND GO TO OUR CALCULATOR.
WE'RE IN DEGREE MODE, SO THAT'S OKAY.
AND THERE'S TANGENT 4/3,
SO THETA IS APPROXIMATELY 53.1 DEGREES
WHICH AGAIN IS OUR DIRECTION ANGLE.
LET'S TALK ABOUT THE HORIZONTAL AND VERTICAL COMPONENTS AGAIN.
THE HORIZONTAL AND VERTICAL COMPONENTS OF A VECTOR U
HAVE A MAGNITUDE OF U
AND DIRECTION THETA ARE GIVEN
BY "A"" IS EQUAL TO THE MAGNITUDE OF U x COSINE THETA
AND B IS EQUAL TO THE MAGNITUDE OF U x SINE THETA.
THEREFORE, IT FOLLOWS THAT WE CAN FIND THE POSITION VECTOR
BY USING THIS FORMULA.
AND AGAIN THESE FORMULAS SHOULD MAKE SENSE,
SHOULD REMIND US OF POLAR COORDINATES.
WE TAKE THE COSINE OF ANGLE THETA,
THAT WOULD BE "A"/THE MAGNITUDE OF U,
SOLVING IT FOR "A,"
WE WOULD HAVE MAGNITUDE OF U x COSINE THETA.
AND SIMILARLY THE SINE OF ANGLE THETA
WOULD BE B DIVIDED BY THE MAGNITUDE OF U
AND THEN WE COULD SOLVE THAT FOR B TO OBTAIN THIS EQUATION HERE.
WE WANT TO FIND THE HORIZONTAL INVERT
FOR COMPONENTS OF VECTOR V THAT HAS A MAGNITUDE 34
AND DIRECTION ANGLE OF 37.2 DEGREES.
LET'S GO AHEAD AND SKETCH THIS AS A POSITION VECTOR.
THIS WOULD BE 37.2 DEGREES WITH MAGNITUDE 34.
SO WE KNOW WE'RE GOING TO BE IN THE FIRST QUADRANT.
THE HORIZONTAL COMPONENT "A" IS EQUAL TO THE MAGNITUDE OF U,
34 x COSINE THETA AND THE VERTICAL COMPONENT = B
WHICH IS 34 x SINE, 37.2 DEGREES.
LET'S GO TO OUR CALCULATOR. WE'RE ALREADY IN DEGREE MODE,
SO THE HORIZONTAL COMPONENT IS APPROXIMATELY 27.1
AND THE VERTICAL COMPONENT WOULD BE APPROXIMATELY 20.6.
AND WE CAN WRITE THIS IN COMPONENT FORM AS FOLLOWS.
A UNIT VECTOR IS A VECTOR WITH MAGNITUDE 1,
2 USEFUL UNIT VECTORS ARE VECTOR "I"
WHICH IS EQUAL TO THE UNIT VECTOR [1,0]
AND VECTOR J WHICH IS EQUAL TO THE UNIT VECTOR [0,1].
WITH UNIT VECTORS "I" AND J
ANY POSITION VECTOR IN THIS FORM CAN BE EXPRESSED
IN THE FORM OF "A" x VECTOR "I" + B x VECTOR J.
SO NOW WE CAN EXPRESS
ANY POSITION VECTOR IN 2 DIFFERENT FORMS.
SO FOR EXAMPLE IF VECTOR V IS EQUAL TO THE VECTOR [5,7],
WE COULD ALSO SAY THAT'S EQUAL
TO 5 x VECTOR "I," + 7 x VECTOR J.
OKAY. PROBABLY ENOUGH FOR THE FIRST VIDEO,
WE'LL GET INTO SOME ADDITIONAL VECTOR OPERATIONS
IN THE NEXT VIDEO.
THANK YOU FOR WATCHING.