Question

Find the coordinates of the points which trisect \[AB\] given that $A(2,1, - 3)$ and $B(5, - 8,3)$.

Answer 1

Hint: We need to understand the condition given in the problem and then we have to use appropriate formulas to find the coordinates of the points which trisect \[AB\]. We have to use the section formula for internal division to calculate the coordinates of the points which trisect \[AB\].
Formula used:
Section formula for internal division:
Coordinates of the point $P(x,y,z)$ which divides line segment joining $A({x_1},{y_1},{z_1})$ and $B({x_2},{y_2},{z_2})$ internally in the ratio $m:n$ are given by,
\[P(x,y,z) = (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}})\]

Complete step-by-step solution:
Let us consider points $P$ and $Q$ trisect \[AB\].
Let us draw the diagram using the above information.

Therefore,
\[AP = PQ = BQ\]
$A(2,1, - 3)$ And $B(5, - 8,3)$
Therefore $P$divides segment \[AB\] in the ratio $1:2$ internally.
Let us apply section formula for internal division,
Coordinates of the point $P(x,y,z)$ are given by,
\[P(x,y,z) = (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}})\]
Let us calculate x-coordinate of $P$,
\[x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}\] \[......[1]\]
$P$Divides segment \[AB\] in the ratio $1:2$ internally.
$A(2,1, - 3)$ And $B(5, - 8,3)$
Therefore, \[m = 1,n = 2,{x_1} = 2,{x_2} = 5\]
Let us put above values in equation \[[1]\],
\[x = \dfrac{{(1)(5) + (2)(2)}}{{1 + 2}}\]
On performing multiplication and additions in the numerator and in the denominator we get,
\[x = \dfrac{9}{3}\]
On performing division we get,
\[x = 3\]
This is the x-coordinate of $P$.
Let us calculate y-coordinate of$P$,
\[y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}\] \[......[2]\]
$P$ Divides segment \[AB\] in the ratio $1:2$ internally.
$A(2,1, - 3)$And$B(5, - 8,3)$
Therefore, \[m = 1,n = 2,{y_1} = 1,{y_2} = - 8\]
Let us put above values in equation \[[2]\],
\[y = \dfrac{{(1)( - 8) + (2)(1)}}{{1 + 2}}\]
On performing multiplication and additions in the numerator and in the denominator we get,
\[y = \dfrac{{ - 6}}{3}\]
On performing division we get,
\[y = - 2\]
This is the y-coordinate of $P$.
Let us calculate z-coordinate of $P$,
\[z = \dfrac{{m{z_2} + n{z_1}}}{{m + n}}\] \[......[3]\]
$P$ Divides segment \[AB\] in the ratio $1:2$ internally.
$A(2,1, - 3)$And$B(5, - 8,3)$
Therefore, \[m = 1,n = 2,{z_1} = - 3,{z_2} = 3\]
Let us put above values in equation\[[3]\],
\[z = \dfrac{{(1)(3) + (2)( - 3)}}{{1 + 2}}\]
On performing multiplication and additions in the numerator and in the denominator we get,
\[z = \dfrac{{ - 3}}{3}\]
On performing division we get,
\[z = - 1\]
This is the z-coordinate of $P$.
Therefore coordinate of point $P$ are \[(3, - 2, - 1)\]
Let, $Q$ divides segment \[BC\]in the ratio $2:1$ internally.
Let us apply section formula for internal division,
Coordinates of the point $Q(x,y,z)$ are given by,
\[Q(x,y,z) = (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}})\]
Let us calculate x-coordinate of$P$,
\[x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}\] \[......[1]\]
$Q$ Divides segment \[BC\] in the ratio $2:1$ internally.
$A(2,1, - 3)$And$B(5, - 8,3)$
Therefore, \[m = 2,n = 1,{x_1} = 2,{x_2} = 5\]
Let us put above values in equation \[[1]\],
\[x = \dfrac{{(2)(5) + (1)(2)}}{{2 + 1}}\]
On performing multiplication and additions in the numerator and in the denominator we get,
\[x = \dfrac{{12}}{3}\]
On performing division we get,
\[x = 4\]
This is the x-coordinate of $Q$.
Let us calculate y-coordinate of $Q$,
\[y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}\] \[......[2]\]
$Q$ Divides segment \[BC\] in the ratio $2:1$ internally.
$A(2,1, - 3)$ And $B(5, - 8,3)$
Therefore, \[m = 2,n = 1,{y_1} = 1,{y_2} = - 8\]
Let us put above values in equation \[[2]\],
\[y = \dfrac{{(2)( - 8) + (1)(1)}}{{2 + 1}}\]
On performing multiplication and additions in the numerator and in the denominator we get,
\[y = \dfrac{{ - 15}}{3}\]
On performing division we get,
\[y = - 5\]
This is the y-coordinate of $Q$.
Let us calculate z-coordinate of $Q$,
\[z = \dfrac{{m{z_2} + n{z_1}}}{{m + n}}\] \[......[3]\]
$Q$ Divides segment \[BC\] in the ratio $2:1$ internally.
$A(2,1, - 3)$ And $B(5, - 8,3)$
Therefore, \[m = 2,n = 1,{z_1} = - 3,{z_2} = 3\]
Let us put above values in equation \[[3]\],
\[z = \dfrac{{(2)(3) + (1)( - 3)}}{{2 + 1}}\]
On performing multiplication and additions in the numerator and in the denominator we get,
\[z = \dfrac{3}{3}\]
On performing division we get,
\[z = 1\]
This is the z-coordinate of $Q$.
Therefore coordinate of point $Q$ are \[(4, - 5,1)\]
Therefore coordinates of the points which trisect \[AB\] are \[(3, - 2, - 1)\] and \[(4, - 5,1)\].

Note: Coordinate of $Q$ can also be calculated using midpoint formula for segment $PB$ , as from the diagram we can see that $Q$ is the midpoint of segment $PB$ and can find the coordinate of P by the midpoint formula after finding the coordinates of point Q as P behaves as the midpoint of AQ. The midpoint formula for 3-D coordinates $\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2},\dfrac{z_1+z_2}{2}\right)$