Simple Nature - Light and Matter

where the vectors ˆx, ŷ, and ẑ, called the unit vectors, are definedas the vectors that have magnitude equal to 1 and directions lyingalong the x, y, and z axes. In speech, they are referred to as “x-hat,”“y-hat,” and “z-hat.”A slightly different, and harder to remember, version of thisnotation is unfortunately more prevalent. In this version, the unitvectors are called î, ĵ, and ˆk:∆r = (290 km)î + (230 km)ĵ .Applications to relative motion, momentum, and forceVector addition is the correct way to generalize the one-dimensionalconcept of adding velocities in relative motion, as shown in the followingexample:Velocity vectors in relative motion example 65⊲ You wish to cross a river and arrive at a dock that is directlyacross from you, but the river’s current will tend to carry youdownstream. To compensate, you must steer the boat at an angle.Find the angle θ, given the magnitude, |v W L |, of the water’svelocity relative to the land, and the maximum speed, |v BW |, ofwhich the boat is capable relative to the water.⊲ The boat’s velocity relative to the land equals the vector sum ofits velocity with respect to the water and the water’s velocity withrespect to the land,v BL = v BW + v W L .If the boat is to travel straight across the river, i.e., along the yaxis, then we need to have v BL,x = 0. This x component equalsthe sum of the x components of the other two vectors,v B L, x = v B W , x + v W L, x ,or0 = −|v BW | sin θ + |v W L | .Solving for θ, we findsin θ = |v W L |/|v BW | ,q / Example 65soθ = sin −1 |v W L |v BW.Section 3.4 Motion In Three Dimensions 201