Let A = (0,0,0), B = (9,8,12), and C = (6,2,3). Find coordinates for the point on line AB that is closest to C.
frog
can someone plz help me?
I got 2.109 instead of 3.253 but thanks for helping me! It must've taken you a long time to type all of that.
Math is intriguing because there are many ways to solve the same problem.
Here's another way by minimizing the distance between the line and the point.
1. find the parametric form of the line AB, which is relatively easy because the line passes through the origin, so the parametric form is
AB: <9t,8t,12t>
(note: <...> represents a vector)
2. find the function that represents the square of the distance of (6,2,3) from AB:
d(t)=(9t-6)^2+(8t-2)^2+(12t-3)^2
3. find the value of t that minimizes the distance d(t):
d(d(t))/dt=24(12t-3)+18(9t-6)+16(8t-2)=0
which simplifies to 2(289t-106)=0
or
t=106/289
4. Substitute t in AB to get
(954/289, 848/289, 1272/289)
5. The distance between C and AB is then the square-root of d(106/289)
=sqrt(2925/289)=3.181
Well, let's see if I can bring some "coordinates" humor to this question!
To find the point on line AB that is closest to C, we can use a fun concept called projections. So, let's "project" ourselves into solving this problem.
First, we need to find the vector representing the line AB. To do that, we subtract the coordinates of point A from the coordinates of point B. Let's call this vector AB.
AB = (9,8,12) - (0,0,0) = (9,8,12)
Now, let's find the vector between point A and C, which we'll call AC.
AC = (6,2,3) - (0,0,0) = (6,2,3)
To find the point on line AB that is closest to C, we can use the dot product. The dot product between AB and AC divided by the squared length of AB gives us the distance from A to that point.
distance = (AB ⋅ AC) / ||AB||^2
Now, let's calculate all the boring numbers:
AB ⋅ AC = (9,8,12) ⋅ (6,2,3) = 9*6 + 8*2 + 12*3 = 54 + 16 + 36 = 106
||AB||^2 = (9^2 + 8^2 + 12^2) = 81 + 64 + 144 = 289
Substituting these values into the distance formula:
distance = 106 / 289 = 0.3668 (rounded)
To find the actual point on line AB, we multiply this distance by the vector AB and add it to point A.
Point on line AB = A + (0.3668) * AB
Point on line AB = (0,0,0) + (0.3668) * (9,8,12)
Point on line AB = (3.302, 2.9384, 4.4056) (rounded)
So, the coordinates of the point on line AB that is closest to C are approximately (3.302, 2.9384, 4.4056).
To find the point on line AB that is closest to C, we need to find the perpendicular distance between C and line AB. Here's how you can do it:
1. Calculate the vector AB. This can be done by subtracting the coordinates of point A from point B.
AB = B - A = (9-0, 8-0, 12-0) = (9, 8, 12)
2. Calculate the vector AC. This can be done by subtracting the coordinates of point A from point C.
AC = C - A = (6-0, 2-0, 3-0) = (6, 2, 3)
3. Calculate the dot product between AB and AC.
dot product = AB · AC = (9 * 6) + (8 * 2) + (12 * 3) = 54 + 16 + 36 = 106
4. Calculate the magnitude of AB.
magnitude of AB = √(9^2 + 8^2 + 12^2) = √(81 + 64 + 144) = √(289) = 17
5. Calculate the scalar projection of AC onto AB.
scalar projection = dot product / magnitude of AB = 106 / 17 = 6.24 (approx)
6. Calculate the closest point P on line AB to C.
P = A + ((scalar projection) * (AB / magnitude of AB))
= (0, 0, 0) + (6.24 / 17) * (9, 8, 12)
≈ (0.408, 0.361, 0.541)
Therefore, the coordinates of the point on line AB closest to C are approximately (0.408, 0.361, 0.541).
The direction of vector AB is (9,8,12)
So a plane perpendicular to vector AB will be
9x + 8y + 12z = c, where c is a constant.
Let's find the plane such that C(6,2,3) lies on it
9(6) + 8(2) + 12(3) = c
c = 106
so the plane containing C and having AB as a normal is
9x + 8y + 12z = 106
the line AB has parametric equation of
x = 0+9t
y=0+8t
z = 0 + 12t
so to find where AB cuts the plane:
9(9t) + 8(8t) + 12(12t) = 106
289t = 106
t = 106/289
So the point of intersection of the plane and the line is
x = 9(106/289) = 954/289
y = 8(106/289) = 848/289
z = 12(106/289) = 1272/289
now use the distance formula to find the shortest distance:
distance
= √(6 - 954/289)^2 + (2 - 848/289)^2 + (3-1272/289)^2 )
= √( (780/289)^2 + (-270/289)^2 + (-450/289)^2)
= (1/289)√883800
= (10/289)√8838 = appr 3.253
check my arithmetic, I was expecting nicer numbers.