Engineering Calculus Notes 85

Engineering Calculus Notes 85 - 1.6. CROSS PRODUCTS 73 →...

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Unformatted text preview: 1.6. CROSS PRODUCTS 73 → → → Recall that any vector − = x− + y − in the plane can also be written in v ı “polar” form as → − = |− | (cos θ − + sin θ − ) → → → v v vı v → where θv is the counterclockwise angle between − and the horizontal v → − . Thus, vector ı θ = θ2 − θ1 − − → → 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− − → − → − → − − → = ( 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 − − → − − → − → 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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