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Second, this result is positive, because the positive sense is defined by the vector system. In this right handed system find the positive rotation by pointing your right hand thumb towards the positive axis (the ‘k’ means that the vector is about the z-axis here), and curl your fingers, that is the positive direction. page 60 ASIDE: The cross (or vector) product of two vectors will yield a new vector perpendicular to both vectors, with a magnitude that is a product of the two magnitudes. V1 × V2 V1 V2 V1 × V2 = ( x 1 i + y1 j + z1 k ) × ( x2 i + y 2 j + z 2 k ) i j k V1 × V2 = x1 y1 z1 x2 y2 z2 V 1 × V 2 = ( y 1 z 2 – z 1 y 2 )i + ( z 1 x 2 – x 1 z 2 )j + ( x 1 y 2 – y 1 x 2 )k ASIDE: The positive orientation of angles and moments about an axis can be determined by pointing the thumb of the right hand along the axis of rotation.

In particular if we want to find the x and y components of F relative to the x-y axis we can use the dot product. λ x = 1i + 0j (unit vector for the x-axis) F x = λ x • F = ( 1i + 0j ) • [ ( 10 cos 60° )i + ( 10 sin 60° )j ] ∴ = ( 1 ) ( 10 cos 60° ) + ( 0 ) ( 10 sin 60° ) = 10N cos 60° This result is obvious, but consider the other obvious case where we want to project a vector onto itself, page 32 10 cos 60°i + 10 sin 60°j F λ F = ------ = --------------------------------------------------------- = cos 60°i + sin 60°j F 10 Incorrect - Not using a unit vector FF = F • F = ( ( 10 cos 60° )i + ( 10 sin 60° )j ) • ( ( 10 cos 60° )i + ( 10 sin 60° )j ) = ( 10 cos 60° ) ( 10 cos 60° ) + ( 10 sin 60° ) ( 10 sin 60° ) 2 2 = 100 ( ( cos 60° ) + ( sin 60° ) ) = 100 Using a unit vector FF = F • λF = ( ( 10 cos 60° )i + ( 10 sin 60° )j ) • ( ( cos 60° )i + ( sin 60° )j ) = ( 10 cos 60° ) ( cos 60° ) + ( 10 sin 60° ) ( sin 60° ) 2 2 = 10 ( ( cos 60° ) + ( sin 60° ) ) = 10 Correct Now consider the case where we find the component of F in the x’ direction.

But, in the case of the pirate on the gangplank, there would be great concern about the plank bending. • The classic example is the see-saw, page 55 10 kg 20 kg M MEDIUM M SMALL The two children sit on the teeter-totter, because they have different masses, they must sit different distances from the centre of rotation, or face catastrophic impact. In mathematical terms the moments on either side of the centre must balance. First, recall the basic equation for a moment. 81 ------- ( 1m ) = 196Nm Kg To solve the problem using proper notation, + ∑M = 0 ∴M SMALL – M MEDIUM = 0 ∴196 – 196 = 0 This shows that the system is static, and the children will balance.

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