Engineering Calculus Notes 121

Engineering Calculus Notes 121 - 1.7 APPLICATIONS OF CROSS...

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Unformatted text preview: 1.7. APPLICATIONS OF CROSS PRODUCTS 109 These new rectangular coordinates can be expressed in terms of the old ones, using the angle-summation formulas for sine and cosine, as x ( α ) = x cos α − y sin α y ( α ) = x sin α + y cos α z ( α ) = z. (1.29) Under a steady rotation around the z-axis with angular velocity 23 ˙ α = ω radians/sec , the velocity −→ v of our point is given by ˙ x = parenleftBig dx ( α ) dα vextendsingle vextendsingle vextendsingle α =0 parenrightBig ω = ( − x sin0 − y cos 0) ω = − yω ˙ y == parenleftBig dy ( α ) dα vextendsingle vextendsingle vextendsingle α =0 parenrightBig ω = ( x cos 0 − y sin0) ω = xω ˙ z = 0 which can also be expressed as −→ v = − yω −→ ı + xω −→ = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle −→ ı −→ −→ k ω x y z vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = ω −→ k × −→ p (1.30)(1....
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