A radar station, located at the origin of xz plane, as shown in the figure , detects an airplane coming straight at the station from the east. At first observation (point A), the position of the airplane relative to the origin is&nbsp;RA→. The position vector&nbsp;RA→&nbsp;has a magnitude of&nbsp;360m&nbsp;and is located at exactly&nbsp;40∘&nbsp;above the horizon. The airplane is tracked for another&nbsp;123∘} in the vertical east-west plane for&nbsp;5.0s, until it has passed directly over the station and reached point B. The position of point B relative to the origin is&nbsp;RB→&nbsp;(the magnitude of&nbsp;RB→&nbsp;is&nbsp;880m). The contact points are shown in the diagram, where the x axis represents the ground and the positive z direction is upward.Define the displacement of the airplane while the radar was tracking it:&nbsp;R→BA=R→B−R→A. What are the components of&nbsp;R→BAExpress&nbsp;R→BA&nbsp;in meters as an ordered pair, separating the x and z components with a comma, to two significant figures.

Question

A radar station, located at the origin of xz plane, as shown in the figure , detects an airplane coming straight at the station from the east. At first observation (point A), the position of the airplane relative to the origin is&amp;nbsp;RA→. The position vector&amp;nbsp;RA→&amp;nbsp;has a magnitude of&amp;nbsp;360m&amp;nbsp;and is located at exactly&amp;nbsp;40∘&amp;nbsp;above the horizon. The airplane is tracked for another&amp;nbsp;123∘} in the vertical east-west plane for&amp;nbsp;5.0s, until it has passed directly over the station and reached point B. The position of point B relative to the origin is&amp;nbsp;RB→&amp;nbsp;(the magnitude of&amp;nbsp;RB→&amp;nbsp;is&amp;nbsp;880m). The contact points are shown in the diagram, where the x axis represents the ground and the positive z direction is upward.Define the displacement of the airplane while the radar was tracking it:&amp;nbsp;R→BA=R→B−R→A. What are the components of&amp;nbsp;R→BAExpress&amp;nbsp;R→BA&amp;nbsp;in meters as an ordered pair, separating the x and z components with a comma, to two significant figures.

sovienesY · Accepted Answer

The position vector RA→ is given by R→A=(rcos⁡θ)i^+(rsin⁡θ)k^ Substitute (360m) for r and 40 deegres for θ to find RA→ R→A=((360m)cos⁡40∘)i^+((360m)sin⁡40∘)k^ =(275.8m)i^+(231.4m)k^ The position vector RB→ is given by, R→B=(rcos⁡θ)i^+(rsin⁡θ)k^ The angle for the position vector over→{R}B along vertical and horizontal plane is, θ=123∘+40∘=163∘ Substitute (880m) for r and 163 deegres for θ to find RB→ R→B=((880m)cos⁡163∘)i^+((880m)sin⁡163∘)k^ =(−841.5m)i^+(257.3m)k^ The resultant displacement vectors components of the air plane are, R→BA=R→B−R→A Substitute (−841.5m)i^+(257.3m)k^ for RB→ and (275.8m)i^+(231.4m)k^ for RA→ to find RBA→. R→BA=(−841.5m)i^+(257.3m)k^−(275.8m)i^+(231.4m)k^ =−(1117.3m)i^+(25.9m)k^ Round off the components of resultant displacement vectors in to two significant figures, R→BA=−(1100m)i^+(26m)k^ Therefore, the displacement of the airplane as an ordered pair separated by the x and z components is (-1100m,26m).

Jeffrey Jordon · Accepted Answer

Vector A:  cos⁡40=x360; x = 276  sin⁡40∘=y360; y = 231  RAx,RAz=(275.775,231.40)  Vector B:  ∠(123+40=163;180−163=17∘)  cos⁡17=x880;  x=−841.55  sin⁡17=y880;  y=257.29  RBx,RBz=(−841.55,257.29)  RBA=B→−A→=(−841.55,257.29)−(275.775,231.40)=(−116.84,25.89)

A radar station, located at the origin of xz plane, as shown in the figure , detects an airplane coming straight at the station from the east.

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