Find the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes $3x-y- 3z + 32 =0$ and $3x+2y-15z= 8$.
Please give me an outline to solve the problem. Thanks.
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asked Apr 15, 2016 at 7:47
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$\begingroup$ What's a "join" of two points? $\endgroup$– 5xumApr 15, 2016 at 7:49
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$\begingroup$ @5xum point on the join of (−3,7,−13) and (−6,1,−10) means a point of the straight line segment joining (−3,7,−13) and (−6,1,−10) $\endgroup$– user1942348Apr 15, 2016 at 7:52
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1 Answer
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You do not tell us what mathematics you already know. Here is one outline, which is not clever but is straightforward. This method avoids calculus, since you did not use that tag, but it does use vectors. Let us know if one of these steps is too difficult for you.
- Find the vector $v_1$ between those two given points $A_0,A_1$ and the parametric equations of the line between them $l_1$.
- Solve the two given equations for the planes and find the parametric equations of the line of intersection $l_2$ and a direction vector $v_2$ for that line.
- Find a vector $v$ perpendicular to the two vectors you just found $v_1,v_2$.
- For each of your two lines, use the perpendicular vector to find the equation of the plane $p_1,p_2$ through the line that contains the vector. Then find the point of intersection $C_1,C_2$ of that plane and the other line.
- See if the point of intersection $C_1$ on your first line is between the two given points $A_0,A_1$. If it is, that is your desired "nearest point." If it is not, find the distance between each of your two given points $A_0,A_1$ and the point $C_2$ on the other line $l_2$. The closer one of $A_0,A_1$ is your desired nearest point.
answered Apr 15, 2016 at 10:09