Determine the least value that the area of the triangle OAP can take. O is the origin, A is known and P is an unknown point on a line

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I have two lines



$$
l_1:
begincases
x=-14-4t\
y=5+t\
z=t
endcases
$$



$$
l_2:
begincases
x=t\
y=2t\
z=2t
endcases
$$



I have a known point A on $l_2$,



$$
A=(1,2,2)
$$



and an unknown point P on $l_1$, which I want to use to minimize the area of the triangle OAP, where O is the origin. How do I do this? I'm having troubles visualizing this as I tried doing it by finding the shortest distance between the lines to use as the length of one of the sides in the triangle and use the formula for triangles $fracbcdot h2$, but I got the wrong answer. So it must be something else.




linear-algebra vector-spaces vectors




share|cite|improve this question



















  • Use area triangle $textOAP=frac12left|oversetrightharpoonup textOA times oversetrightharpoonup textOPright|$ where "$times $" is the cross product
    – Lozenges
    Jul 28 at 14:53











  • How would I find P?
    – Chisq
    Jul 28 at 14:54










  • $P(-14-4 t,5+t,t)$ is an arbitrary point on $l_1$. Write the area as a function of $t$ and minimize. Are we allowed to use Calculus?
    – Lozenges
    Jul 28 at 15:13










  • Hmm. Not if that entails any derivatives or anything. What did you have in mind?
    – Chisq
    Jul 28 at 15:16










  • Hmm. How about using the shortest distance as the height $PH$. It is not one of the sides.
    – Lozenges
    Jul 28 at 15:22














up vote
1
down vote

favorite












I have two lines



$$
l_1:
begincases
x=-14-4t\
y=5+t\
z=t
endcases
$$



$$
l_2:
begincases
x=t\
y=2t\
z=2t
endcases
$$



I have a known point A on $l_2$,



$$
A=(1,2,2)
$$



and an unknown point P on $l_1$, which I want to use to minimize the area of the triangle OAP, where O is the origin. How do I do this? I'm having troubles visualizing this as I tried doing it by finding the shortest distance between the lines to use as the length of one of the sides in the triangle and use the formula for triangles $fracbcdot h2$, but I got the wrong answer. So it must be something else.




linear-algebra vector-spaces vectors




share|cite|improve this question



















  • Use area triangle $textOAP=frac12left|oversetrightharpoonup textOA times oversetrightharpoonup textOPright|$ where "$times $" is the cross product
    – Lozenges
    Jul 28 at 14:53











  • How would I find P?
    – Chisq
    Jul 28 at 14:54










  • $P(-14-4 t,5+t,t)$ is an arbitrary point on $l_1$. Write the area as a function of $t$ and minimize. Are we allowed to use Calculus?
    – Lozenges
    Jul 28 at 15:13










  • Hmm. Not if that entails any derivatives or anything. What did you have in mind?
    – Chisq
    Jul 28 at 15:16










  • Hmm. How about using the shortest distance as the height $PH$. It is not one of the sides.
    – Lozenges
    Jul 28 at 15:22












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I have two lines



$$
l_1:
begincases
x=-14-4t\
y=5+t\
z=t
endcases
$$



$$
l_2:
begincases
x=t\
y=2t\
z=2t
endcases
$$



I have a known point A on $l_2$,



$$
A=(1,2,2)
$$



and an unknown point P on $l_1$, which I want to use to minimize the area of the triangle OAP, where O is the origin. How do I do this? I'm having troubles visualizing this as I tried doing it by finding the shortest distance between the lines to use as the length of one of the sides in the triangle and use the formula for triangles $fracbcdot h2$, but I got the wrong answer. So it must be something else.




linear-algebra vector-spaces vectors




share|cite|improve this question











I have two lines



$$
l_1:
begincases
x=-14-4t\
y=5+t\
z=t
endcases
$$



$$
l_2:
begincases
x=t\
y=2t\
z=2t
endcases
$$



I have a known point A on $l_2$,



$$
A=(1,2,2)
$$



and an unknown point P on $l_1$, which I want to use to minimize the area of the triangle OAP, where O is the origin. How do I do this? I'm having troubles visualizing this as I tried doing it by finding the shortest distance between the lines to use as the length of one of the sides in the triangle and use the formula for triangles $fracbcdot h2$, but I got the wrong answer. So it must be something else.





linear-algebra vector-spaces vectors





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 28 at 14:21









Chisq

1025




1025











  • Use area triangle $textOAP=frac12left|oversetrightharpoonup textOA times oversetrightharpoonup textOPright|$ where "$times $" is the cross product
    – Lozenges
    Jul 28 at 14:53











  • How would I find P?
    – Chisq
    Jul 28 at 14:54










  • $P(-14-4 t,5+t,t)$ is an arbitrary point on $l_1$. Write the area as a function of $t$ and minimize. Are we allowed to use Calculus?
    – Lozenges
    Jul 28 at 15:13










  • Hmm. Not if that entails any derivatives or anything. What did you have in mind?
    – Chisq
    Jul 28 at 15:16










  • Hmm. How about using the shortest distance as the height $PH$. It is not one of the sides.
    – Lozenges
    Jul 28 at 15:22
















  • Use area triangle $textOAP=frac12left|oversetrightharpoonup textOA times oversetrightharpoonup textOPright|$ where "$times $" is the cross product
    – Lozenges
    Jul 28 at 14:53











  • How would I find P?
    – Chisq
    Jul 28 at 14:54










  • $P(-14-4 t,5+t,t)$ is an arbitrary point on $l_1$. Write the area as a function of $t$ and minimize. Are we allowed to use Calculus?
    – Lozenges
    Jul 28 at 15:13










  • Hmm. Not if that entails any derivatives or anything. What did you have in mind?
    – Chisq
    Jul 28 at 15:16










  • Hmm. How about using the shortest distance as the height $PH$. It is not one of the sides.
    – Lozenges
    Jul 28 at 15:22















Use area triangle $textOAP=frac12left|oversetrightharpoonup textOA times oversetrightharpoonup textOPright|$ where "$times $" is the cross product
– Lozenges
Jul 28 at 14:53





Use area triangle $textOAP=frac12left|oversetrightharpoonup textOA times oversetrightharpoonup textOPright|$ where "$times $" is the cross product
– Lozenges
Jul 28 at 14:53













How would I find P?
– Chisq
Jul 28 at 14:54




How would I find P?
– Chisq
Jul 28 at 14:54












$P(-14-4 t,5+t,t)$ is an arbitrary point on $l_1$. Write the area as a function of $t$ and minimize. Are we allowed to use Calculus?
– Lozenges
Jul 28 at 15:13




$P(-14-4 t,5+t,t)$ is an arbitrary point on $l_1$. Write the area as a function of $t$ and minimize. Are we allowed to use Calculus?
– Lozenges
Jul 28 at 15:13












Hmm. Not if that entails any derivatives or anything. What did you have in mind?
– Chisq
Jul 28 at 15:16




Hmm. Not if that entails any derivatives or anything. What did you have in mind?
– Chisq
Jul 28 at 15:16












Hmm. How about using the shortest distance as the height $PH$. It is not one of the sides.
– Lozenges
Jul 28 at 15:22




Hmm. How about using the shortest distance as the height $PH$. It is not one of the sides.
– Lozenges
Jul 28 at 15:22















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