Find the exact coordinates of all possible points D on the line through A and B so that D is four times as...
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0
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A(4, 7, -3)
B(-3, 1, 2)
AB <-7, -6, 5>
parametric equation for AB: x = 4 - 7t ; y = 7 - 6t ; z = -3 + 5t
I tried to use the distance formula where I set 4d (d being the distance of D to B) as the distance from D to A. I really don't know where to a) go from there because I got stuck or b) begin.
The question says "exact coordinates" so I would assume there are multiple coordinates that fit the criteria. I guess this means I'd have to make/find a general equation to find all the points but I don't know where to start on that either.
linear-algebra multivariable-calculus 3d
asked Dec 4 at 0:21
ufotink
43
|
show 1 more comment
up vote
0
down vote
favorite
A(4, 7, -3)
B(-3, 1, 2)
AB <-7, -6, 5>
parametric equation for AB: x = 4 - 7t ; y = 7 - 6t ; z = -3 + 5t
I tried to use the distance formula where I set 4d (d being the distance of D to B) as the distance from D to A. I really don't know where to a) go from there because I got stuck or b) begin.
The question says "exact coordinates" so I would assume there are multiple coordinates that fit the criteria. I guess this means I'd have to make/find a general equation to find all the points but I don't know where to start on that either.
linear-algebra multivariable-calculus 3d
asked Dec 4 at 0:21
ufotink
43
In your title, you have “... as it ($D$) is from $D$.” Presumably, that last $D$ is meant to be a $B$.
– amd
Dec 4 at 3:50
Can you write down a parametric equation for the line?
– amd
Dec 4 at 3:53
@amd yes, I know how to write the parametric equation if that's what you're asking. I edited the question to include the parametric equation
– ufotink
Dec 4 at 4:46
In that case you’re almost done. Work out the distances to $A$ and $B$, set up the equation that expresses the distance constraint, and solve for $t$.
– amd
Dec 4 at 5:24
that doesn't make sense
– ufotink
Dec 4 at 5:45
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A(4, 7, -3)
B(-3, 1, 2)
AB <-7, -6, 5>
parametric equation for AB: x = 4 - 7t ; y = 7 - 6t ; z = -3 + 5t
I tried to use the distance formula where I set 4d (d being the distance of D to B) as the distance from D to A. I really don't know where to a) go from there because I got stuck or b) begin.
The question says "exact coordinates" so I would assume there are multiple coordinates that fit the criteria. I guess this means I'd have to make/find a general equation to find all the points but I don't know where to start on that either.
linear-algebra multivariable-calculus 3d
asked Dec 4 at 0:21
ufotink
43
A(4, 7, -3)
B(-3, 1, 2)
AB <-7, -6, 5>
parametric equation for AB: x = 4 - 7t ; y = 7 - 6t ; z = -3 + 5t
I tried to use the distance formula where I set 4d (d being the distance of D to B) as the distance from D to A. I really don't know where to a) go from there because I got stuck or b) begin.
The question says "exact coordinates" so I would assume there are multiple coordinates that fit the criteria. I guess this means I'd have to make/find a general equation to find all the points but I don't know where to start on that either.
linear-algebra multivariable-calculus 3d
linear-algebra multivariable-calculus 3d
asked Dec 4 at 0:21
ufotink
43
asked Dec 4 at 0:21
ufotink
43
edited Dec 4 at 4:45
asked Dec 4 at 0:21
ufotink
43
asked Dec 4 at 0:21
ufotink
43
asked Dec 4 at 0:21
ufotink
43
43
In your title, you have “... as it ($D$) is from $D$.” Presumably, that last $D$ is meant to be a $B$.
– amd
Dec 4 at 3:50
Can you write down a parametric equation for the line?
– amd
Dec 4 at 3:53
@amd yes, I know how to write the parametric equation if that's what you're asking. I edited the question to include the parametric equation
– ufotink
Dec 4 at 4:46
In that case you’re almost done. Work out the distances to $A$ and $B$, set up the equation that expresses the distance constraint, and solve for $t$.
– amd
Dec 4 at 5:24
that doesn't make sense
– ufotink
Dec 4 at 5:45
|
show 1 more comment
In your title, you have “... as it ($D$) is from $D$.” Presumably, that last $D$ is meant to be a $B$.
– amd
Dec 4 at 3:50
Can you write down a parametric equation for the line?
– amd
Dec 4 at 3:53
@amd yes, I know how to write the parametric equation if that's what you're asking. I edited the question to include the parametric equation
– ufotink
Dec 4 at 4:46
In that case you’re almost done. Work out the distances to $A$ and $B$, set up the equation that expresses the distance constraint, and solve for $t$.
– amd
Dec 4 at 5:24
that doesn't make sense
– ufotink
Dec 4 at 5:45
In your title, you have “... as it ($D$) is from $D$.” Presumably, that last $D$ is meant to be a $B$.
– amd
Dec 4 at 3:50
In your title, you have “... as it ($D$) is from $D$.” Presumably, that last $D$ is meant to be a $B$.
– amd
Dec 4 at 3:50
Can you write down a parametric equation for the line?
– amd
Dec 4 at 3:53
Can you write down a parametric equation for the line?
– amd
Dec 4 at 3:53
@amd yes, I know how to write the parametric equation if that's what you're asking. I edited the question to include the parametric equation
– ufotink
Dec 4 at 4:46
@amd yes, I know how to write the parametric equation if that's what you're asking. I edited the question to include the parametric equation
– ufotink
Dec 4 at 4:46
In that case you’re almost done. Work out the distances to $A$ and $B$, set up the equation that expresses the distance constraint, and solve for $t$.
– amd
Dec 4 at 5:24
In that case you’re almost done. Work out the distances to $A$ and $B$, set up the equation that expresses the distance constraint, and solve for $t$.
– amd
Dec 4 at 5:24
that doesn't make sense
– ufotink
Dec 4 at 5:45
that doesn't make sense
– ufotink
Dec 4 at 5:45
|
show 1 more comment
2 Answers
2
active
oldest
votes
up vote
0
down vote
Hint:
what is the point in
$$
begin{pmatrix}
x\y\z
end{pmatrix}=
begin{pmatrix}
4\7\-3
end{pmatrix}
+tbegin{pmatrix}
-7\-6\5
end{pmatrix}
$$
for $t=frac{4}{5}$ ?
answered Dec 4 at 9:52
Emilio Novati
51.2k43472
add a comment |
up vote
0
down vote
There is a formula in coordinate geometry to find the coordinates of a point between 2 points $(x_1, y_1, z_1) and (x_2, y_2, z_3)$ in the ratio of m:n.
$$x = frac{nx_1+mx_2}{m+n}$$
$$y = frac{ny_1+my_2}{m+n}$$
$$z = frac{nz_1+mz_2}{m+n}$$
There are 2 cases
Case 1 : D is between A and B
Apply the above formula we have D equals $(frac{-8}{5}, frac{11}{5}, 1)$
Case 2 : D is closer to B on AB's extension, then treat B as the middle point and AB:BD = 3:1. We have
$frac{3x+4}{4} = -3$ $x=frac{-16}{3}$
Similarly y=-1 and z=11/3.
answered Dec 4 at 16:02
KY Tang
12
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hint:
what is the point in
$$
begin{pmatrix}
x\y\z
end{pmatrix}=
begin{pmatrix}
4\7\-3
end{pmatrix}
+tbegin{pmatrix}
-7\-6\5
end{pmatrix}
$$
for $t=frac{4}{5}$ ?
answered Dec 4 at 9:52
Emilio Novati
51.2k43472
add a comment |
up vote
0
down vote
Hint:
what is the point in
$$
begin{pmatrix}
x\y\z
end{pmatrix}=
begin{pmatrix}
4\7\-3
end{pmatrix}
+tbegin{pmatrix}
-7\-6\5
end{pmatrix}
$$
for $t=frac{4}{5}$ ?
answered Dec 4 at 9:52
Emilio Novati
51.2k43472
add a comment |
up vote
0
down vote
up vote
0
down vote
Hint:
what is the point in
$$
begin{pmatrix}
x\y\z
end{pmatrix}=
begin{pmatrix}
4\7\-3
end{pmatrix}
+tbegin{pmatrix}
-7\-6\5
end{pmatrix}
$$
for $t=frac{4}{5}$ ?
answered Dec 4 at 9:52
Emilio Novati
51.2k43472
Hint:
what is the point in
$$
begin{pmatrix}
x\y\z
end{pmatrix}=
begin{pmatrix}
4\7\-3
end{pmatrix}
+tbegin{pmatrix}
-7\-6\5
end{pmatrix}
$$
for $t=frac{4}{5}$ ?
answered Dec 4 at 9:52
Emilio Novati
51.2k43472
answered Dec 4 at 9:52
Emilio Novati
51.2k43472
answered Dec 4 at 9:52
Emilio Novati
51.2k43472
answered Dec 4 at 9:52
Emilio Novati
51.2k43472
51.2k43472
add a comment |
add a comment |
up vote
0
down vote
There is a formula in coordinate geometry to find the coordinates of a point between 2 points $(x_1, y_1, z_1) and (x_2, y_2, z_3)$ in the ratio of m:n.
$$x = frac{nx_1+mx_2}{m+n}$$
$$y = frac{ny_1+my_2}{m+n}$$
$$z = frac{nz_1+mz_2}{m+n}$$
There are 2 cases
Case 1 : D is between A and B
Apply the above formula we have D equals $(frac{-8}{5}, frac{11}{5}, 1)$
Case 2 : D is closer to B on AB's extension, then treat B as the middle point and AB:BD = 3:1. We have
$frac{3x+4}{4} = -3$ $x=frac{-16}{3}$
Similarly y=-1 and z=11/3.
answered Dec 4 at 16:02
KY Tang
12
add a comment |
up vote
0
down vote
There is a formula in coordinate geometry to find the coordinates of a point between 2 points $(x_1, y_1, z_1) and (x_2, y_2, z_3)$ in the ratio of m:n.
$$x = frac{nx_1+mx_2}{m+n}$$
$$y = frac{ny_1+my_2}{m+n}$$
$$z = frac{nz_1+mz_2}{m+n}$$
There are 2 cases
Case 1 : D is between A and B
Apply the above formula we have D equals $(frac{-8}{5}, frac{11}{5}, 1)$
Case 2 : D is closer to B on AB's extension, then treat B as the middle point and AB:BD = 3:1. We have
$frac{3x+4}{4} = -3$ $x=frac{-16}{3}$
Similarly y=-1 and z=11/3.
answered Dec 4 at 16:02
KY Tang
12
add a comment |
up vote
0
down vote
up vote
0
down vote
There is a formula in coordinate geometry to find the coordinates of a point between 2 points $(x_1, y_1, z_1) and (x_2, y_2, z_3)$ in the ratio of m:n.
$$x = frac{nx_1+mx_2}{m+n}$$
$$y = frac{ny_1+my_2}{m+n}$$
$$z = frac{nz_1+mz_2}{m+n}$$
There are 2 cases
Case 1 : D is between A and B
Apply the above formula we have D equals $(frac{-8}{5}, frac{11}{5}, 1)$
Case 2 : D is closer to B on AB's extension, then treat B as the middle point and AB:BD = 3:1. We have
$frac{3x+4}{4} = -3$ $x=frac{-16}{3}$
Similarly y=-1 and z=11/3.
answered Dec 4 at 16:02
KY Tang
12
There is a formula in coordinate geometry to find the coordinates of a point between 2 points $(x_1, y_1, z_1) and (x_2, y_2, z_3)$ in the ratio of m:n.
$$x = frac{nx_1+mx_2}{m+n}$$
$$y = frac{ny_1+my_2}{m+n}$$
$$z = frac{nz_1+mz_2}{m+n}$$
There are 2 cases
Case 1 : D is between A and B
Apply the above formula we have D equals $(frac{-8}{5}, frac{11}{5}, 1)$
Case 2 : D is closer to B on AB's extension, then treat B as the middle point and AB:BD = 3:1. We have
$frac{3x+4}{4} = -3$ $x=frac{-16}{3}$
Similarly y=-1 and z=11/3.
answered Dec 4 at 16:02
KY Tang
12
answered Dec 4 at 16:02
KY Tang
12
answered Dec 4 at 16:02
KY Tang
12
answered Dec 4 at 16:02
KY Tang
12
12
add a comment |
add a comment |
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In your title, you have “... as it ($D$) is from $D$.” Presumably, that last $D$ is meant to be a $B$.
– amd
Dec 4 at 3:50
Can you write down a parametric equation for the line?
– amd
Dec 4 at 3:53
@amd yes, I know how to write the parametric equation if that's what you're asking. I edited the question to include the parametric equation
– ufotink
Dec 4 at 4:46
In that case you’re almost done. Work out the distances to $A$ and $B$, set up the equation that expresses the distance constraint, and solve for $t$.
– amd
Dec 4 at 5:24
that doesn't make sense
– ufotink
Dec 4 at 5:45