Simple Nature - Light and Matter

A shortcut example 62⊲ A split second after nine o’clock, the hour hand on a clock dialhas moved clockwise past the nine-o’clock position by some imperceptiblysmall angle φ. Let positive x be to the right and positivey up. If the hand, with length l, is represented by a ∆r vectorgoing from the dial’s center to the tip of the hand, find this vector’s∆x.⊲ The following shortcut is the easiest way to work out exampleslike these, in which a vector’s direction is known relative to oneof the axes. We can tell that ∆r will have a large, negative xcomponent and a small, positive y. Since ∆x < 0, there arereally only two logical possibilities: either ∆x = −l cos φ, or ∆x =−l sin φ. Because φ is small, cos φ is large and sin φ is small.We conclude that ∆x = −l cos φ.A typical application of this technique to force vectors is given inexample 71 on p. 205.Addition of vectors given their componentsThe easiest type of vector addition is when you are in possessionof the components, and want to find the components of their sum.San Diego to Las Vegas example 63⊲ Given the ∆x and ∆y values from the previous examples, findthe ∆x and ∆y from San Diego to Las Vegas.⊲∆x total = ∆x 1 + ∆x 2= −120 km + 290 km= 170 kmn / Example 63.∆y total = ∆y 1 + ∆y 2= 150 km + 230 km= 380Addition of vectors given their magnitudes and directionsIn this case, you must first translate the magnitudes and directionsinto components, and the add the components.Graphical addition of vectorsOften the easiest way to add vectors is by making a scale drawingon a piece of paper. This is known as graphical addition, as opposedto the analytic techniques discussed previously.From San Diego to Las Vegas, graphically example 64⊲ Given the magnitudes and angles of the ∆r vectors from SanDiego to Los Angeles and from Los Angeles to Las Vegas, findthe magnitude and angle of the ∆r vector from San Diego to LasSection 3.4 Motion In Three Dimensions 199