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vectors

# vectors - can be found from a = a x 2 a y 2 a z 2 = a •...

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V ECTORS Reference : Any undergraduate text in dynamics. We will deal with the following types of vectors in this course: position vectors, velocity vectors, acceleration vectors and force vectors. For the most part, we will express these vectors in terms of their Cartesian (xyz) components. (a) This xyz coordinate system will be described unit vectors of i , j , and k . For example, the force F shown below can be written as: F = F x i + F y j + F z k x y z F F z F x F y i j k (b) The xyz coordinates will be based on a “right-handed coordinate system”. That is, if you sweep the fingers of your RIGHT hand from the unit vector i to the unit vector j , your thumb will point in the direction of the unit vector k .

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(c) The “dot (scalar) product” of two vectors a and b is found by: a b = a x i + a y j + a z k ( 29 b x i + b y j + b z k ( 29 = a x b x i i + a x b y i j + a x b z i k + a y b x j i + a y b y j j + a y b z j k + a z b x k i + a z b y k j + a z b z k k = a x b x + a y b y + a z b z since i • j = i • k = j • k = 0 and i • i = j • j = k • k = 1. (d) From above, we see that the magnitude of a vector a
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Unformatted text preview: can be found from: a = a x 2 + a y 2 + a z 2 = a • a (e) The “cross (vector) product” of two vectors a and b is found by: a × b = a x i + a y j + a z k ( 29 × b x i + b y j + b z k ( 29 = a x b x i × i + a x b y i × j + a x b z i × k + a y b x j × i + a y b y j × j + a y b z j × k + a z b x k × i + a z b y k × j + a z b z k × k = a x b x + a x b y k + a x b z-j ( 29 + a y b x-k ( 29 + a y b y + a y b z i + a z b x j + a z b y-i ( 29 + a z b z = a y b z-a z b y [ ] i + a z b x-a x b z [ ] j + a x b y-a y b x [ ] k since i × i = j × j = k × k = i × j = -j × i = k i × k = -k × i = j j × k = -k × j = i Note that the above results depends on having a right-handed coordinate system....
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