Vector problem

[Joesuss]

Hi there, Could anyone please help me with this problem? I am having difficulty solving it

Given the line L : X = (9, 13, -3) + t(1, 4, -2); t ∈ R, the point A with position vector
4i +16j -3k and that the point P lies on L such that AP is perpendicular to L, find the exact
coordinates of P.

Thanks in Advance
Joe

 

[Deleted member 4993]

Hi there, Could anyone please help me with this problem? I am having difficulty solving it

Given the line L : X = (9, 13, -3) + t(1, 4, -2); t ∈ R, the point A with position vector
4i +16j -3k and that the point P lies on L such that AP is perpendicular to L, find the exact
coordinates of P.

Thanks in Advance
Joe

If I were to do this problem - I would start with a sketch of the problem. Have you done that?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[pka]

Given the line L : X = (9, 13, -3) + t(1, 4, -2); t ∈ R, the point A with position vector
4i +16j -3k and that the point P lies on L such that AP is perpendicular to L, find the exact coordinates of P.

The vector \(\overrightarrow {AP} =\left<p-4,q-16,r+3\right>\) where \(P: (p,q,r)\).
\(\overrightarrow {AP}\cdot<1,4,-2> =\bf{0}\)
Also it must be true that \((\exists t_0)[p=9+t_0, q=13+4t_0, r=-3-2t_0]\)
Can you finish?

 

[Joesuss]

Hi, Here's what i have done so far. Doesn't look right as i cant imagine there being 2 irrational answers to the co-ordinates of P.

 

[Joesuss]

Does this look better? Still unsure :/

 

[pka]

To Joesuss, I have made a personal decision: you have received help which is freely given.
That help is given by some very highly educated people.
Therefore I will no longer try to read posted images.
They are simply a way for the lazy poster to avoid posting in a readable format.
You can learn to post in LaTeX code. That way any of us professionals can read it.
It is up to you, do you need truly professional help?
If not please ignore this post.;

 

[HallsofIvy]

The fact that "A has position vector 4i +16j -3k simply means that A has coordinates (x, y, z)= (4, 16, -3). If P lies on the given line then its (x, y, z) coordinates satisfy the equation (x, y, z)= (9, 13, -3) + t(1, 4, -2)= (9+ t. 13+ 4t, -3- 2t). The vector AP has components (4- (9+ t))i+ (16- (13+ 4t))j+ (-3-(-3-2t))k= (-5+ t)i+ (3- 4t)j- 2tk. That vector is to be perpendicular to the line so perpendicular to the "direction vector" i+ 4j- 2k (the vector coefficient of the variable t). So their dot product must be 0: (-5+ t)+ 4(3- 4t)- 2(-2t)= -5+ t+ 12- 16t+ 4t= 7- 11t= 0. t= 7/11 so what is P?