Find the coordinates of a point in 3-space

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The discussion revolves around finding the coordinates of a point B in 3-space, given a vector A and the distance between points A and B. The participants analyze the vector representation of B, utilizing unit vectors and magnitudes to establish relationships between the coordinates. They derive a quadratic equation to solve for the magnitude of vector B, ultimately identifying the correct value needed to determine the coordinates of point B. The conversation highlights the importance of careful algebraic manipulation and verification of each step in the calculations. The final goal is to accurately locate point B based on the established conditions.
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Homework Statement
The vector from the origin to point ##A## is given as ##(6,-2,-4)##, and the unit vector directed from the origin toward point ##B## is ##(2,-2,1)/3##. If points ##A## and ##B## are ten units apart, find the coordinates of point ##B##.
Relevant Equations
##\left |\vec r\right |=10##
##\hat a_B=\frac {\vec B}{\left |\vec B\right |}={\frac {\vec B}{\sqrt {B^2_x+B^2_y +B^2_z}}}##
##\left| \vec r\right |=\sqrt {(x_B-x_A)^2+(y_B-y_A)^2+(z_B-z_A)^2}##
Ans:##(7.8,-7.8,3.9)##
##\hat a_B=\frac 2 3\hat a_x-\frac 2 3\hat a_y+\frac 1 3\hat a_z##
##\left| \vec r\right |=\sqrt {(x_B-6)^2+(y_B+2)^2+(z_B+4)^2}=10##
##\left |\vec A\right |=\sqrt {6^2+2^2+4^2}=\sqrt {5}{6}##

Not sure where to go from here. Please help!

Source: Problem 1.3; Engineering Electromagnetics, 8th Edition, William Hayt, John Buck
 
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Hi,
can you write an expression for the vector from A to B?
Edit: ah, your ##\vec r##.
Rewrite those x,y,z as ##|\vec B| \hat a_B##
 
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You are given that
##\vec A=\{6,-2,-4\}##
##\vec B=B\{\frac{2}{3},-\frac{2}{3},\frac{1}{3}\}##
##|\vec B-\vec A|=10## units
Does this help?
 
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kuruman said:
##B\{\frac{2}{3},-\frac{2}{3},\frac{1}{3}\}##
Why does this expression have a ##B## out front? Didn't it already get distributed inside?
 
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CaliforniaRoll88 said:
Why does this expression have a ##B## out front? Didn't it already get distrubuted inside?
Nothing is distributed.

A vector ##\vec B## can always be written as its magnitude ##B## (a scalar) times a unit vector ##\hat b## in the same direction as ##\vec B##, i.e. ##\vec B=B~\hat b.## Then $$\vec B\cdot\vec B=(B~\hat b)\cdot(B~\hat b)=B^2(\hat b\cdot\hat b)=B^2(1)=B^2$$ as it should. Here ##\hat b=\{\frac{2}{3},-\frac{2}{3},\frac{1}{3}\}.## You can dot it with itself and verify that it is a unit vector.
 
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BvU said:
Rewrite those x,y,z as ##|\vec B| \hat a_B##
##\vec B=|\vec B|\hat a_B=|\vec B|\frac 2 3\hat a_x-|\vec B|\frac 2 3\hat a_y+|\vec B|\frac 1 3\hat a_z##
 
Look at the last equation in post #3. Do you see that this equation has only one unknown? What is this unknown and how will its value help you find the answer to this problem?
 
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##\left ||\vec B|\hat a_B-\vec A\right |=10##
 
kuruman said:
Look at the last equation in post #3. Do you see that this equation has only one unknown? What is this unknown and how will its value help you find the answer to this problem?
The unknown is ##|\vec B|##. If I find out this value I can scale my unit vector to point B.
Is ##|\vec B|=3##?
 
  • #10
CaliforniaRoll88 said:
The unknown is ##|\vec B|##. If I find out this value I can scale my unit vector to point B.
Is ##|\vec B|=3##?
Right. Go for it.
 
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  • #11
CaliforniaRoll88 said:
##\left |\vec A\right |=\sqrt {6^2+2^2+4^2}=\sqrt {5}{6}##
Just a note on the LaTeX… you meant ##\sqrt {56}##.
 
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  • #12
haruspex said:
Just a note on the LaTeX… you meant ##\sqrt {56}##.
I did, thanks.
 
  • #13
kuruman said:
Right. Go for it.
##\vec B=|\vec B|\hat a_B=3\hat a_B=2\hat a_x-2\hat a_y+\hat a_z##
##B(2,-2,1)##
This isn't the answer.
 
  • #14
CaliforniaRoll88 said:
##\vec B=|\vec B|\hat a_B=3\hat a_B=2\hat a_x-2\hat a_y+\hat a_z##
##B(2,-2,1)##
This isn't the answer.
I am not surprised that this is not the answer. How did you establish that ##|\vec B|=3## units? Just because there is a number ##3## in the denominator? A vector ##\vec C## that is a scalar number ##N## times ##\vec B## has the same unit vector and is written as ##\vec C=C\{\frac{2}{3},-\frac{2}{3},\frac{1}{3}\}.## Would you follow the same reasoning and say that ##|\vec C|=3## units?

Use the last equation in post #3.
 
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  • #15
kuruman said:
Use the last equation in post #3.
##|\vec B-\vec A|=\left ||\vec B|\hat a_B+(-\vec A)\right |=10##
##\left< |\vec B|\frac {2}{3}-6,-|\vec B|\frac {2}{3}+2, |\vec B|\frac {1}{3}+4\right >##
I am stuck at this point. Please help!
 
  • #16
What is an expression for the magnitude of a vector if you know its 3 components? Write it down and set it equal to 10 units. While you're at it, do yourself a favor and simplify from ##|\vec B|## to just ##B##. It will make the algebra look less daunting.
 
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  • #17
kuruman said:
What is an expression for the magnitude of a vector if you know its 3 components?
##\sqrt {(B_x\frac {2}{3}-6)^2+(-B_y\frac {2}{3}+2)^2+(B_z\frac {1}{3}+4)^2}=10##
At this point it seems that I have 3 variables and 1 equation. Any hints of how to proceed?
 
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  • #18
The middle term under the radical doesn't look right.

After you correct it, can you solve the equation for ##B##?
 
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  • #19
CaliforniaRoll88 said:
##\sqrt {(B_x\frac {2}{3}-6)^2+(-B_y\frac {2}{3}+2)^2+(B_z\frac {1}{3}+4)^2}=10##
At this point it seems that I have 3 variables and 1 equation. Any hints of how to proceed?
You have 3 variables because you introduced them when you edited the post which you shouldn't have done. Anyway, let's backtrack.
##|\vec B-\vec A|=\sqrt{(B_x-A_x)^2+(B_y-A_y)^2+(B_z-A_z)^2}.##

##B_x=\dfrac{2}{3}B##; ##B_y=-\dfrac{2}{3}B##; ##B_z=\dfrac{1}{3}B##.

##-A_x=-6##; ##-A_y=+2##; ##-A_z=+4##.

Substitute.
 
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  • #20
kuruman said:
Substitute.

##\sqrt {(B\frac {2}{3}-6)^2+(-B\frac {2}{3}+2)^2+(B\frac {1}{3}+4)^2}=10##
##B^2-\frac {28}{3}B-32=0##
Using the Quadratic Formula: ##B=12,-2\frac{2}{3}##.
Therefore, discarding the negative value and using ##B=12## I get ##B(2,-6,8)## which is wrong.
Do you know where I went wrong? Thank you.
 
  • #21
CaliforniaRoll88 said:
##\sqrt {(B\frac {2}{3}-6)^2+(-B\frac {2}{3}+2)^2+(B\frac {1}{3}+4)^2}=10##
##B^2-\frac {28}{3}B-32=0##
Using the Quadratic Formula: ##B=12,-2\frac{2}{3}##.
Therefore, discarding the negative value and using ##B=12## I get ##B(2,-6,8)## which is wrong.
Do you know where I went wrong? Thank you.
You show the first and and last equations and nothing else. Since I don't know what you did in-between, all I can say is that you went wrong somewhere between the first equation and the last equation. The solutions of the quadratic are correct.
 
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  • #22
kuruman said:
You show the first and and last equations and nothing else. Since I don't know what you did in-between, all I can say is that you went wrong somewhere between the first equation and the last equation.
##\sqrt {(B\frac {2}{3}-6)^2+(-B\frac {2}{3}+2)^2+(B\frac {1}{3}+4)^2}=10##
##\left(B\frac{2}{3}-6\right)^2=\frac {4}{9}B^2-2(B\frac {2}{3})(6)+(-6)^2=\frac {4}{9}B^2-\frac{24}{3}B+36##
##\left(-B\frac{2}{3}+2\right)^2=\frac {4}{9}B^2-2(2)(B\frac {2}{3})+(4)^2=\frac {4}{9}B^2-\frac{8}{3}B+16##
##\left(B\frac{1}{3}+4\right)^2=\frac {1}{9}B^2-2(B\frac {1}{3})(4)+(4)^2=\frac {1}{9}B^2-\frac{4}{3}B+16##
##\frac {4}{9}B^2+\frac {4}{9}B^2+\frac {1}{9}B^2=B^2##
##-\frac {24}{3}B-\frac {8}{3}B+\frac {4}{3}B=-\frac {28}{3}B##
##36+16+16=68##
##B^2-\frac {28}{3}B+68=100##
##B^2-\frac {28}{3}B-32=0##
##B=12##
##B\frac {2}{3}-6=12*\frac {2}{3}-6=2##
##-B\frac {2}{3}+2=-12*\frac {2}{3}+2=-6##
##B\frac {1}{3}+4=12*\frac {1}{3}+4=8##
##\left<2,-6,8\right>##
Do you see where I went wrong now?
 
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  • #23
kuruman said:
You are given that
##\vec A=\{6,-2,-4\}##
##\vec B=B\{\frac{2}{3},-\frac{2}{3},\frac{1}{3}\}##
##|\vec B-\vec A|=10## units
Does this help?
Note that the factor of ##\frac 1 3## is an unnecessary complication. Instead, we can look for:
$$\vec B=B\{2, -2, 1\}$$
 
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  • #24
CaliforniaRoll88 said:
##\left(-B\frac{2}{3}+2\right)^2=\frac {4}{9}B^2-2(2)(B\frac {2}{3})+(4)^2=\frac {4}{9}B^2-\frac{8}{3}B+16##
This is wrong! The ##2## became ##4##.
 
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  • #25
An alternative approach is to use the scalar product of vectors. Suppose we have two vectors ##\vec a## and ##\vec b## and we want the distance from the point A (##\vec a##) and B (##B\vec b##) to be some number ##c##.

First, we have:$$\vec a \cdot \vec b = ab \cos \theta$$Then, we have the law of cosines:$$c^2 = a^2 + B^2b^2 - 2Bab \cos \theta = a^2 + B^2b^2 - 2B \vec a \cdot \vec b$$That should give the same quadratic equation for ##B##.

And, you can take ##\vec b## to be a unit vector or not, depending on what looks simpler.
 
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  • #26
CaliforniaRoll88 said:
##\left(B\frac{1}{3}+4\right)^2=\frac {1}{9}B^2-2(B\frac {1}{3})(4)+(4)^2=\frac {1}{9}B^2-\frac{4}{3}B+16##
Try again.
CaliforniaRoll88 said:
##\left(-B\frac{2}{3}+2\right)^2=\frac {4}{9}B^2-2(2)(B\frac {2}{3})+(4)^2=\frac {4}{9}B^2-\frac{8}{3}B+16##
As @PeroK said, This is also wrong.
 
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  • #27
@CaliforniaRoll88, to reduce the chance of algebraic and arithmetic mistakes, it helps if you can simplify the working.

##\vec B = |\vec B| ~\frac 13<2, -2, 1>##

Let ##k = \frac 13 |\vec B|## (e.g. see @PeroK, Post #23).

##\vec B = <2k, -2k, k>## and the coordinates of point B are ##(2k, -2k, k)##. Solve for k.
 
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  • #28
CaliforniaRoll88 said:
##\sqrt {(B\frac {2}{3}-6)^2+(-B\frac {2}{3}+2)^2+(B\frac {1}{3}+4)^2}=10##
##\left(B\frac{2}{3}-6\right)^2=\frac {4}{9}B^2-2(B\frac {2}{3})(6)+(-6)^2=\frac {4}{9}B^2-\frac{24}{3}B+36##
##\left(-B\frac{2}{3}+2\right)^2=\frac {4}{9}B^2-2(2)(B\frac {2}{3})+(4)^2=\frac {4}{9}B^2-\frac{8}{3}B+16##
##\left(B\frac{1}{3}+4\right)^2=\frac {1}{9}B^2-2(B\frac {1}{3})(4)+(4)^2=\frac {1}{9}B^2-\frac{4}{3}B+16##
##\frac {4}{9}B^2+\frac {4}{9}B^2+\frac {1}{9}B^2=B^2##
##-\frac {24}{3}B-\frac {8}{3}B+\frac {4}{3}B=-\frac {28}{3}B##
##36+16+16=68##
##B^2-\frac {28}{3}B+68=100##
##B^2-\frac {28}{3}B-32=0##
##B=12##
##B\frac {2}{3}-6=12*\frac {2}{3}-6=2##
##-B\frac {2}{3}+2=-12*\frac {2}{3}+2=-6##
##B\frac {1}{3}+4=12*\frac {1}{3}+4=8##
##\left<2,-6,8\right>##
Do you see where I went wrong now?
Thanks for posting the details of your derivation. Yes, I do see where you went wrong. You subtracted the coordinates of the tip of ##\vec A## twice. Once when you found ##|\vec B-\vec A|## and then again at the very end after you found ##B##.

Once you have found the length of ##\vec B##, point B that you are looking for is at the tip of the arrow representing vector ##\vec B.~## If ## \vec B =12\{\frac{2}{3},-\frac{2}{3},\frac{1}{3}\}##, where is its tip?

Conceptually, you can think of reaching the solution as follows:
  1. Construct given vector ##\vec A##.
  2. At the tip of ##\vec A## construct a sphere of radius ##|\vec B-\vec A| = 10## units. Point B that you are looking for must lie on this sphere.
  3. Draw unit vector ##\{\frac{2}{3},-\frac{2}{3},\frac{1}{3}\}## then extend it until it intersects the sphere. The point of intersection is what you are looking for.
Note that if you connect the ends of ##\vec A## and ##\vec B## with a straight line, you form a triangle to which you can apply the rule of cosines as suggested by @PeroK in post #25.
 
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  • #29
There is another error here
##\left(B\frac{1}{3}+4\right)^2=\frac {1}{9}B^2-2(B\frac {1}{3})(4)+(4)^2=\frac {1}{9}B^2-\frac{4}{3}B+16.##
The sign in front of the linear term in ##B## should be ##+## not ##-##.
 
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  • #30
MatinSAR said:
Try again.
kuruman said:
There is another error here
##\left(B\frac{1}{3}+4\right)^2=\frac {1}{9}B^2-2(B\frac {1}{3})(4)+(4)^2=\frac {1}{9}B^2-\frac{4}{3}B+16.##
The sign in front of the linear term in ##B## should be ##+## not ##-##.
##\left(B\frac{2}{3}-6\right)^2=\frac {4}{9}B^2-2(B\frac {2}{3})(6)+(-6)^2=\frac {4}{9}B^2-\frac{24}{3}B+36##
##\left(\frac{1}{3}B+4\right)^2=(\frac {1}{3}B)^2+2(\frac {1}{3}B)(4)+(4)^2=\frac {1}{9}B^2+\frac{4}{3}B+16##
##\left(-\frac{2}{3}B+2\right)^2=(\frac {2}{3}B)^2+2(\frac {2}{3}B)(2)+(2)^2=\frac {4}{9}B^2+\frac{8}{3}B+4##
##\frac {4}{9}B^2+\frac {4}{9}B^2+\frac {1}{9}B^2=B^2##
##-\frac {24}{3}B+\frac {8}{3}B+\frac {4}{3}B=-\frac {12}{3}B=4B##
##36+16+4=56##
##B^2-4B+56=100##
##B^2-4B-44=0##
##x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a}##
##B = \frac {-(-4) \pm \sqrt{(-4)^2 -4(1)(-44)}} {2(1)}##
##B = \frac {4 \pm \sqrt{16 +176}} {2(1)}##
##B = \frac {4 \pm \sqrt{192}} {2}##
 
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  • #31
CaliforniaRoll88 said:
##\left(B\frac{2}{3}-6\right)^2=\frac {4}{9}B^2-2(B\frac {2}{3})(6)+(-6)^2=\frac {4}{9}B^2-\frac{24}{3}B+36##
##\left(\frac{1}{3}B+4\right)^2=(\frac {1}{3}B)^2+2(\frac {1}{3}B)(4)+(4)^2=\frac {1}{9}B^2+\frac{4}{3}B+16##
##\left(-\frac{2}{3}B+2\right)^2=(\frac {2}{3}B)^2+2(\frac {2}{3}B)(2)+(2)^2=\frac {4}{9}B^2+\frac{8}{3}B+4##
##\frac {4}{9}B^2+\frac {4}{9}B^2+\frac {1}{9}B^2=B^2##
##-\frac {24}{3}B+\frac {8}{3}B+\frac {4}{3}B=-\frac {12}{3}B=4B##
##36+16+4=56##
##B^2-4B+56=100##
##B^2-4B-44=0##
##x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a}##
##B = \frac {-(-4) \pm \sqrt{(-4)^2 -4(1)(-44)}} {2(1)}##
##B = \frac {4 \pm \sqrt{16 +176}} {2(1)}##
##B = \frac {4 \pm \sqrt{192}} {2}##
##\left(\frac{1}{3}B+4\right)^2=(\frac {1}{3}B)^2+\color{red}{2(\frac {1}{3}B)(4)}+(4)^2=\frac {1}{9}B^2+\color{red}{\frac{4}{3}B}+16##
 
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  • #32
There is another one that escaped my previous pass.
##\left(-\frac{2}{3}B+2\right)^2=(\frac {2}{3}B)^2\color{red}{+}2(\frac {2}{3}B)(2)+(2)^2=\frac {4}{9}B^2+\frac{8}{3}B+4##

I suggest that you redo and verify each equation before you typeset it in LaTeX.
 
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  • #33
kuruman said:
##\left(\frac{1}{3}B+4\right)^2=(\frac {1}{3}B)^2+\color{red}{2(\frac {1}{3}B)(4)}+(4)^2=\frac {1}{9}B^2+\color{red}{\frac{4}{3}B}+16##
I think it should be ##\frac 8 3 B##.
 
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  • #34
CaliforniaRoll88 said:
##\left(B\frac{2}{3}-6\right)^2=\frac {4}{9}B^2-2(B\frac {2}{3})(6)+(-6)^2=\frac {4}{9}B^2-\frac{24}{3}B+36##
##\left(\frac{1}{3}B+4\right)^2=(\frac {1}{3}B)^2+2(\frac {1}{3}B)(4)+(4)^2=\frac {1}{9}B^2+\frac{4}{3}B+16##
##\left(-\frac{2}{3}B+2\right)^2=(\frac {2}{3}B)^2+2(\frac {2}{3}B)(2)+(2)^2=\frac {4}{9}B^2+\frac{8}{3}B+4##
##\frac {4}{9}B^2+\frac {4}{9}B^2+\frac {1}{9}B^2=B^2##
##-\frac {24}{3}B+\frac {8}{3}B+\frac {4}{3}B=-\frac {12}{3}B=4B##
##36+16+4=56##
##B^2-4B+56=100##
##B^2-4B-44=0##
##x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a}##
##B = \frac {-(-4) \pm \sqrt{(-4)^2 -4(1)(-44)}} {2(1)}##
##B = \frac {4 \pm \sqrt{16 +176}} {2(1)}##
##B = \frac {4 \pm \sqrt{192}} {2}##
If this was an exam, you would lose a lot of points for no particular reason.
1683152390883.png

You can use https://www.symbolab.com/ to check your calculations.
Edit :
Link
 
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  • #35
##\left(B\frac{2}{3}-6\right)^2=\frac {4}{9}B^2-2(B\frac {2}{3})(6)+(-6)^2=\frac {4}{9}B^2-\frac{24}{3}B+36##
##\left(\frac{1}{3}B+4\right)^2=(\frac {1}{3}B)^2+2(\frac {1}{3}B)(4)+(4)^2=\frac {1}{9}B^2+\frac{8}{3}B+16##
##\left(-\frac{2}{3}B+2\right)^2=(-\frac {2}{3}B)^2+2(-\frac {2}{3}B)(2)+(2)^2=\frac {4}{9}B^2-\frac{8}{3}B+4##
##\frac {4}{9}B^2+\frac {4}{9}B^2+\frac {1}{9}B^2=B^2##
##-\frac {24}{3}B+\frac {8}{3}B-\frac {8}{3}B=-\frac {24}{3}B=-8B##
##36+16+4=56##
##B^2-8B+56=100##
##B^2-8B-44=0##
##x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a}##
##B = \frac {-(-8) \pm \sqrt{(-8)^2 -4(1)(-44)}} {2(1)}##
##B = \frac {8 \pm \sqrt{64+176}}{2}##
##B = \frac {8 \pm \sqrt{240}}{2}##
##B = \frac {8 \pm \sqrt{15*16}}{2}##
##B = \frac {8 \pm 4\sqrt{15}}{2}=##
##B = 4 \pm 2\sqrt{15}=11.74597,-3.74597##
##\vec B =11.74597\{\frac{2}{3},-\frac{2}{3},\frac{1}{3}\}##
##B(7.83,-7.83,3.915)##
Finally got the answer with this method.
 
  • #36
PeroK said:
Note that the factor of ##\frac 1 3## is an unnecessary complication. Instead, we can look for:
$$\vec B=B\{2, -2, 1\}$$
How do you disregard the ##1/3##?
 
  • #37
CaliforniaRoll88 said:
How do you disregard the ##1/3##?
Why do you include it?
 
  • #38
CaliforniaRoll88 said:
How do you disregard the ##1/3##?
Let's look at two questions. Let A be any point:

a) What are the coordinates of a point a distance ##c## units from ##A## in the direction of the vector ##\vec b##?

b) What are the coordinates of a point a distance ##c## units from ##A## in the direction of the vector ##\hat b##?
 
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  • #39
CaliforniaRoll88 said:
Finally got the answer with this method.
It can be done much more easily using the Post #27 and @PeroK's suggestions!
 
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  • #40
PeroK said:
a) What are the coordinates of a point a distance ##c## units from ##A## in the direction of the ##\vec b##?
Let ##B## be any point not ##A##:
Then the coordinates of ##B## away from ##A## by ##c## units is ##B(cA_x,cA_y,cA_z)##
Is this right?
 
  • #41
CaliforniaRoll88 said:
Let ##B## be any point not ##A##:
Then the coordinates of ##B## away from ##A## by ##c## units is ##B(cA_x,cA_y,cA_z)##
Is this right?
That's not right at all. The point of my question was that the answers to a) and b) are the same. A direction is a direction.
 
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  • #42
CaliforniaRoll88 said:
Let ##B## be any point not ##A##:
Then the coordinates of ##B## away from ##A## by ##c## units is ##B(cA_x,cA_y,cA_z)##
Is this right?
I suggest that you do some 2D problems so that you can draw a diagram. I think you are struggling with 3D geometry because you are struggling with the basic concepts.
 
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  • #43
CaliforniaRoll88 said:
Let ##B## be any point not ##A##:
Then the coordinates of ##B## away from ##A## by ##c## units is ##B(cA_x,cA_y,cA_z)##
Is this right?
Hi @CaliforniaRoll88. That's not right. As suggested by @PeroK, maybe it would help you to do a 2D problem first - a diagram is then easy to draw so you can see what's happening. How about trying this:

"The vector from the origin to point ##P## is given as ##<1, 2>##, and the unit vector directed from the origin towards point ##Q## is ##{\frac 15}{<3,4>}## . If points ##P## and ##Q## are ten units apart, find the coordinates of point ##Q##.

And you might want to re-read Post #27!

Edited.
 
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  • #44
Steve4Physics said:
Hi @CaliforniaRoll88. That's not right. As suggested by @PeroK, maybe it would help you to do a 2D problem first - a diagram is then easy to draw so you can see what's happening. How about trying this:

"The vector from the origin to point ##P## is given as ##<1, 2>##, and the unit vector directed from the origin towards point ##Q## is ##{\frac 15}{<3,4>}## . If points ##P## and ##Q## are ten units apart, find the coordinates of point ##Q##.

And you might want to re-read Post #27!

Edited.
##\vec P=P\left<1,2\right>##
##\vec Q=Q\frac{1}{5}\left<3,4\right>##
Let ##k=\frac{1}{5}Q##
##\vec Q=k\left<3,4\right>=\left<3k,4k\right>##
##\vec P\cdot\vec Q=1\cdot3k+2\cdot4k=11k##
##\vec R## is the resultant of ##\vec P## and ##\vec Q##
Law of Cosines:
##R^2=P^2+Q^2-2\vec P\cdot\vec Q##
##10^2=5+25k-22k##
Am I headed in the right direction?
 
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  • #45
CaliforniaRoll88 said:
##\vec P=P\left<1,2\right>##
##\vec Q=Q\frac{1}{5}\left<3,4\right>##
Let ##k=\frac{1}{5}Q##
##\vec Q=k\left<3,4\right>=\left<3k,4k\right>##
##\vec P\cdot\vec Q=1\cdot3k+2\cdot4k=11k##
##\vec R## is the resultant of ##\vec P## and ##\vec Q##
Law of Cosines:
##R^2=P^2+Q^2-2\vec P\cdot\vec Q##
##10^2=5+25k-22k##
Am I headed in the right direction?
Looks good!

PS except, should have a quadratic in ##k## with a ##25k^2## term. I missed that.
 
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  • #46
CaliforniaRoll88 said:
##\vec R## is the resultant of ##\vec P## and ##\vec Q##
A couple of points (pun intended).

1) We want to make ##|\vec {PQ}|=10##. Knowing the resultant ##\vec P + \vec Q## doesn't help. It's possible that you might be thinking ##\vec {PQ} = \vec P + \vec Q## but that's wrong! A suitable diagram will make this clear.

2) There is (IMO) a much simpler approach. The distance between points P and Q must be 10. You are given point P(1, 2). What are coordinates of point Q in terms of k? The rest is simple!
 
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  • #47
##10^2=5+25k-22k##
##95=3k##
##k=\frac {95}{3}##
##\vec Q=k\left<3,4\right>=\left<3k,4k\right>=\left<3\frac {95}{3},4\frac {95}{3}\right>=\left<95,126\frac {2}{3}\right>##
Is this right @Steve4Physics, @PeroK?
Edit: I am sure it's wrong.
 
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  • #48
CaliforniaRoll88 said:
##10^2=5+25k-22k##
##95=3k##
##k=\frac {95}{3}##
##\vec Q=k\left<3,4\right>=\left<3k,4k\right>=\left<3\frac {95}{3},4\frac {95}{3}\right>=\left<95,126\frac {2}{3}\right>##
Is this right @Steve4Physics, @PeroK?
Edit: I am sure it's wrong.
I missed that you had ##25k## instead of ##25k^2##. You need to find some way of reducing the number of simple algebraic errors. And, you need a way of spotting them yourself.
 
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  • #49
CaliforniaRoll88 said:
##\vec P=P\left<1,2\right>##
As a completely separate correction, it’s probably worth noting that the above is wrong. ##\vec P = \left< 1, 2 \right>##. There is no "##P##" on the right-hand side.

The magnitude of ##\vec P## is ##P = \sqrt {1^2 +2^2} = \sqrt 5##. By writing “##P\left< 1, 2 \right>##” you are multiplying ##\vec P## by ##\sqrt 5##; this gives a new vector ##\sqrt 5## times bigger than ##\vec P##.
 
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  • #50
CaliforniaRoll88 said:
##10^2=5+25k-22k##
##95=3k##
##k=\frac {95}{3}##
##\vec Q=k\left<3,4\right>=\left<3k,4k\right>=\left<3\frac {95}{3},4\frac {95}{3}\right>=\left<95,126\frac {2}{3}\right>##
Is this right @Steve4Physics, @PeroK?
Edit: I am sure it's wrong.
Hi @CaliforniaRoll88. Yes, it's wrong!

You seem to be struggling and we are not looking at the original homework problem. So maybe a bit of extra help is justifiable.

Hopefully by now you have drawn a clear diagram (xy axes) with points and vectors marked, estimating the approximate position of point Q ‘by eye’.

You have correctly written (in Post #44): ##\vec Q=k\left<3,4\right>=\left<3k,4k\right>##

##\vec Q## is an ‘arrow’ from the origin to point Q. Therefore the coordinates of point Q are ##(3k, 4k)## where ##k## is some as yet unknown value.

You have 2 points P(1,2) and Q(3k, 4k). You require the distance between them to be 10. So what equation can you now write down and solve?
 
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