Review. You are standing on the ground at the origin of a...

Review. You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the $x$ axis and at a fixed height of $7.60 \times$ $10^{3} \mathrm{~m}$. At time $t=0,$ the airplane is directly above you so that the vector leading from you to it is $\overrightarrow{\mathbf{P}}_{0}=7.60 \times 10^{3} \hat{\mathbf{j}} \mathrm{m}$ At $t=30.0 \mathrm{~s},$ the position vector leading from you to the airplane is $\overrightarrow{\mathbf{P}}_{30}=\left(8.04 \times 10^{3} \hat{\mathbf{i}}+7.60 \times 10^{3} \mathbf{j}\right) \mathrm{m}$ as suggested in Figure $\mathrm{P} 3.31 .$ Determine the magnitude and orientation of the airplane's position vector at $t=45.0 \mathrm{~s}$.

Review. You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the $x$ axis and at a fixed height of $7.60 \times$ $10^{3} \mathrm{~m}$. At time $t=0,$ the airplane is directly above you so that the vector leading from you to it is $\overrightarrow{\mathbf{P}}_{0}=7.60 \times 10^{3} \hat{\mathbf{j}} \mathrm{m}$ At $t=30.0 \mathrm{~s},$ the position vector leading from you to the airplane is $\overrightarrow{\mathbf{P}}_{30}=\left(8.04 \times 10^{3} \hat{\mathbf{i}}+7.60 \times 10^{3} \mathbf{j}\right) \mathrm{m}$ as suggested in
Figure $\mathrm{P} 3.31 .$ Determine the magnitude and orientation of the airplane's position vector at $t=45.0 \mathrm{~s}$.

Key Concepts

Computing Magnitude and Orientation

The magnitude of a vector is determined using the Pythagorean theorem applied to its components, while the orientation (or direction) is found using trigonometric functions, typically by taking the arctangent of the ratio of the vertical to horizontal components. This process converts the component form into a magnitude and an angle relative to one of the axes, providing a complete description of the vector.

Components of Vectors

Breaking vectors into their components allows a multi-dimensional problem to be handled as separate one-dimensional problems. Each vector is expressed in terms of its contributions along the coordinate axes, enabling the use of algebraic methods to compute the overall displacement, magnitude, and direction.

Position Vector

A position vector is a directed line segment that represents the location of a point in space relative to an origin. In a Cartesian coordinate system, it is expressed in terms of its components along the fundamental unit vectors, typically i (horizontal) and j (vertical), which simplifies the analysis of motion in each direction independently.

Uniform Motion

Uniform motion refers to movement at a constant velocity, meaning that the displacement of an object is directly proportional to the elapsed time. When an object moves with uniform motion, its position at any given time can be determined by adding the product of its constant velocity and time interval to its initial position vector.

Vector Addition

Vector addition is the process of combining vectors by adding their corresponding components to obtain a resultant vector. This concept is critical when calculating the overall displacement of an object moving in multiple steps or directions, where each segment of the movement is represented as a vector.

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