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- WE'RE ASKED TO WRITE VECTOR W AS A COMBINATION OF VECTOR U
AND VECTOR V.
TO DO THIS WE'LL FIRST WRITE EACH VECTOR IN COMPONENT FORM.
SO FOR VECTOR U,
THIS WOULD BE THE HORIZONTAL COMPONENT,
AND THIS WOULD BE THE VERTICAL COMPONENT.
SO IF THIS IS X AND THIS IS Y, X WOULD BE -1 AND Y WOULD BE +2.
SO WE CAN WRITE VECTOR U IN COMPONENT FORM AS (-1, 2).
AND THEN FOR VECTOR V THIS WOULD BE THE HORIZONTAL COMPONENT,
AND THE THIS WOULD BE THE VERTICAL COMPONENT.
SO, AGAIN, IF THIS IS X AND THIS Y, X WOULD BE +1, Y WOULD BE +3.
SO VECTOR V COULD BE WRITTEN IN COMPONENT FORM AS (1, 3).
AND THEN FINALLY VECTOR W,
THIS IS THE HORIZONTAL COMPONENT,
THIS IS THE VERTICAL COMPONENT.
SO IF THIS IS X AND THIS IS Y, X IS +1
AND Y IS 1, 2, 3, 4, 5, 6, 7, -7.
SO IF WE WANT TO WRITE VECTOR W
AS A COMBINATION OF VECTOR U AND V,
THEN IF "A" AND B ARE SCALARS,
"A" x VECTOR U + B x VECTOR V = VECTOR W.
SO IN COMPONENT FORM WE WOULD HAVE "A" x VECTOR U
+ B x VECTOR V = VECTOR W.
SO IF I PERFORM THIS MULTIPLICATION,
WE WOULD HAVE -1 x "A," OR JUST (-A, 2A)
+ HERE WE WOULD HAVE (B, 3B).
THIS STILL EQUALS VECTOR W.
SO IF WE ADD THE HORIZONTAL COMPONENTS, IT MUST EQUAL +1.
IF WE ADD THE VERTICAL COMPONENTS,
IT MUST EQUAL -7.
SO THIS GIVES US A SYSTEM OF EQUATIONS TO SOLVE.
WE WOULD HAVE -A + B = +1 AND 2A + 3B = -7.
LET'S GO AHEAD AND SOLVE THIS SYSTEM OF EQUATIONS
ON THE NEXT SLIDE.
LET'S GO AHEAD AND SOLVE THIS FIRST EQUATION FOR "B."
WE'LL ADD "A" TO BOTH SIDES OF THE EQUATION.
SO THIS WOULD GIVE US "B" = 1 + A,
WHICH MEANS WE CAN SUBSTITUTE A + 1 FOR "B"
IN THE SECOND EQUATION.
SO THAT WOULD GIVE US 2A + 3 x B, BUT B = 1 + A = -7.
LET'S GO AHEAD AND SOLVE FOR A.
WE HAVE 2A + 3 + 3A = -7.
COMBINING LIKE TERMS, WE HAVE 5A + 3 = -7.
SUBTRACT 3 ON BOTH SIDES.
DIVIDE BY 5.
SO WE HAVE A = -2.
AND SINCE WE ALSO KNOW THAT B = 1 + A, B = 1 + -2.
SO B = -1.
SO NOW LET'S GO BACK TO OUR PREVIOUS SLIDE.
IF A = -2, AND B = -1, USING OUR EQUATION HERE,
WE CAN SAY THAT VECTOR W = -2 x VECTOR U + -1 x VECTOR V,
OR JUST -VECTOR V.
NOW THAT WE HAVE VECTOR W AS A COMBINATION OF VECTOR U
AND VECTOR V,
LET'S LOOK AT THIS GEOMETRICALLY, AS WELL.
OUR GOAL WAS TO WRITE VECTOR W AS A COMBINATION OF VECTOR U
AND VECTOR V.
AND WE FOUND THAT -2 x VECTOR U - VECTOR V = VECTOR W.
WELL, HERE IS -2 x VECTOR U.
NOTICE HOW IT'S IN THE OPPOSITE DIRECTION OF VECTOR U
AND HAS TWICE THE MAGNITUDE.
AND HERE'S THE OPPOSITE OF VECTOR V, OR -1 x VECTOR V,
AND NOTICE HOW IT HAS THE SAME MAGNITUDE AS VECTOR V,
BUT IT'S IN THE OPPOSITE DIRECTION.
AND THEN, NOTICE WHEN WE SUM THESE TWO VECTORS,
IT IS EQUAL TO VECTOR W.
SO I HOPE THE GRAPH OF THE VECTORS
HELPS EXPLAIN WHAT WE JUST FOUND.