What is the distance of closest approach

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In summary: Minimizing" the square minimizes the distance itself.In summary, the given problem involves finding the distance of closest approach between a point and a line in space. The formula for calculating this distance is given as $d(P,L) = \frac{|\vec{Pr_0} \times \vec{A} |}{|\vec{A}|}$, where $P$ is the point, $L$ is the line, and $r(t) = \vec{r_0} + \vec{A} t$ is the path equation. To find the minimum distance, the distance formula is used and can be minimized by setting the derivative equal to 0 or by completing the square.
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from problem I find \[ r = r_0 + At \] \[ x_0 = 3 + 2t\] \[ y_0 = -1 - 2t\] \[ z_0 = 1 + t\] and \[ A = (2,-2,1)\]
but i don't understand What is the distance of closest approach?
someone tell me to a formula please.
 
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If $P$ is a point in space, and $L$ is the line $r(t) = \vec{r_0} + \vec{A} t$ , then

$d(P,L) = \dfrac{|\vec{Pr_0} \times \vec{A} |}{|\vec{A}|}$
 
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from problem I find \[ r = r_0 + At \] \[ x_0 = 3 + 2t\] \[ y_0 = -1 - 2t\] \[ z_0 = 1 + t\] and \[ A = (2,-2,1)\]

you haven't actually written the path in the form r= r0+ At. It is, of course, r= <3, -1, 1>+ <2, -2, 1>t.
but i don't understand What is the distance of closest approach?

someone tell me to a formula please.

It is the shortest possible distance from a point, (x(t), y(t), z(r)), on the graph to the origin, (0, 0, 0). The "formula" you want is the distance formula. The distance from (x, y, z) to the origin is $\sqrt{x^2+ y^2+ z^2}$. "Minimizing" that distance is the same as minimizing the square- $x^2+ y^2+ z^2= (3+ 2t)^2+ (-1- 2t)^2+ (1+ t)^2= 9+ 12t+ 4t^2+ 1+ 4t+ 4t^2+ 1+ 2t+ t^2= 11+ 18t+ 10t^2$. You can minimize that by setting the derivative equal to 0 or by "completing the square".
To complete the square, $10t^2+ 18t+ 11= 10(t^2+ 1.8t)+ 11= 10(t^2+ 1.8t+ 0.81- 0.81)+ 11= 10(t+ .9)^2- 8.1+ 11= 10(t+ .9)^2+ 2.9. Since a square is never negative, that is smallest when t= -0.9 where its value is 2.9.
 
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FAQ: What is the distance of closest approach

1. What is the distance of closest approach?

The distance of closest approach is the shortest distance between two objects during their interaction or movement. It is often used in physics and astronomy to describe the closest distance between two celestial bodies.

2. How is the distance of closest approach calculated?

The distance of closest approach is calculated using mathematical equations that take into account the initial positions and velocities of the two objects, as well as any forces or interactions between them. These calculations can vary depending on the specific scenario and the type of objects involved.

3. Why is the distance of closest approach important?

The distance of closest approach is important because it can provide valuable information about the nature of the interaction between two objects. It can also help determine the potential for collisions or other significant events, especially in the context of space exploration and satellite orbits.

4. Can the distance of closest approach change over time?

Yes, the distance of closest approach can change over time. This can be due to various factors such as the objects' changing positions and velocities, the influence of external forces, or the objects' changing masses. In some cases, the distance of closest approach may also be affected by the objects' physical properties, such as their size or shape.

5. How is the distance of closest approach related to impact parameter?

The impact parameter is the perpendicular distance between the path of an object and a point of reference, such as the center of another object. The distance of closest approach is related to the impact parameter in that it represents the minimum value of the impact parameter during an interaction between two objects. In other words, the distance of closest approach is the closest the objects will get to each other during their interaction.

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