Unformatted text preview: 1.6. CROSS PRODUCTS 73 →
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Recall that any vector − = x− + y − in the plane can also be written in
v
ı “polar” form as
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− = −  (cos θ − + sin θ − )
→
→
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v
v
vı
v
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where θv is the counterclockwise angle between − and the horizontal
v
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− . Thus,
vector ı
θ = θ2 − θ1
−
−
→
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where θ1 and θ2 are the angles between − and each of the vectors AB ,
ı
−
→
AC , and
θ2 > θ1 .
But the formula for the sine of a sum of angles gives us
sin θ = cos θ1 sin θ2 − cos θ2 sin θ1 .
Thus, if θAC > θAB we have
−−
→→
1−
AB AC sin θ
2
−−
→→
1−
= AB AC (cos θAB sin θAC − cos θAC sin θAB )
2
1−
−
→
−
→
−
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−
−
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=
( AB cos θAB )( AC sin θAC ) − ( AC cos θAC )( AB sin θAB )
2
1
= [xAB yAC − xAC yAB ].
2 A (△ABC ) = −
→
The condition θAC > θAB means that the direction of AC is a
counterclockwise rotation (by an angle between 0 and π radians) from that
−
−
→
−
−
→
−
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of AB ; if the rotation from AB to AC is clockwise, then the two vectors
trade places—or equivalently, the expression above gives us minus the area
of △ABC .
The expression in brackets is easier to remember using a “visual” notation.
An array of four numbers
x1 y 1
x2 y 2
in two horizontal rows, with the entries vertically aligned in columns, is
called a 2 × 2 matrix12 . The determinant of a 2 × 2 matrix is the
product x1 y2 of the downward diagonal minus the product x2 y1 of the
12 pronounced “two by two matrix” ...
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 Calculus, Angles, Dot Product, Euclidean vector, AB AC sin, ϴv, cos θAC sin

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