Answer

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Hint: To solve this problem, we would need the basic concepts of three dimensional geometry and vectors. We further, use the formula for a point to divide line segment that is $\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\dfrac{m{{z}_{2}}+n{{z}_{1}}}{m+n} \right)$, if point divides the line segment in ratio m:n.

Complete step by step solution:

We first find the line segment AB given by A (1,2,3) and B (−3,4,−5). Now, we are to find the ratio in which the XY – plane divides AB. Now, we assume that the XZ−plane divides line AB at point C (x,y,z) in the ratio k:1. Now, the formula for a point C (x,y,z) to divide line segment joining point A ( ${{x}_{1}},{{y}_{1}},{{z}_{1}}$ ) and B( ${{x}_{2}},{{y}_{2}},{{z}_{2}}$ ) in the ratio m:n is given by-

= $\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\dfrac{m{{z}_{2}}+n{{z}_{1}}}{m+n} \right)$

Here, in this case, we are given that the ratio as k:1. Thus, we will use the above formula with m = k and n = 1. Now, we have,

A ( ${{x}_{1}},{{y}_{1}},{{z}_{1}}$ ) = (1,2,3)

B ( ${{x}_{2}},{{y}_{2}},{{z}_{2}}$ ) = (−3,4,−5)

Now, using the above formula, we can find the coordinate of C (x,y,z) as -

C (x, y, z) = $\left( \dfrac{k{{x}_{2}}+{{x}_{1}}}{k+1},\dfrac{k{{y}_{2}}+{{y}_{1}}}{k+1},\dfrac{k{{z}_{2}}+{{z}_{1}}}{k+1} \right)$

(Since, m = k and n = 1)

Now, putting values of points A ( ${{x}_{1}},{{y}_{1}},{{z}_{1}}$ ) and B( ${{x}_{2}},{{y}_{2}},{{z}_{2}}$ ) as (1, 2, 3) and (-3, 4, -5). Thus, we have,

= $\left( \dfrac{k(-3)+1}{k+1},\dfrac{k(4)+2}{k+1},\dfrac{k(-5)+3}{k+1} \right)$ -- (1)

Since point C (x,y,z) lies on the XY – plane, the z coordinate of C will be zero. Thus, we have,

-5k + 3 = 0

k = $\dfrac{3}{5}$

We can substitute the value of k in (1), we get,

= $\left( \dfrac{-0.8}{1.6},\dfrac{4.4}{1.6},0 \right)$

= (-0.5, 2.75, 0) -- (2)

Thus, the required ratio is k:1, that is 0.6 : 1. Simplifying the ratio, we get,

0.6 : 1 = 3 : 5. Thus, the required ratio is 3 : 5.

We get the positive vector of the point of division from (2). Thus, the vector is given by -0.5i + 2.75j. Here, i denotes unit vector to X-axis and j denotes unit vector to Y-axis.

Note: Generally, while solving the problems related to finding points dividing a line segment in a particular ratio, we always try to use the formula by inserting ratio as k : 1, instead of m:n. This way, we only have to deal with one variable, which makes it easier to deal with the obtained equations.

Complete step by step solution:

We first find the line segment AB given by A (1,2,3) and B (−3,4,−5). Now, we are to find the ratio in which the XY – plane divides AB. Now, we assume that the XZ−plane divides line AB at point C (x,y,z) in the ratio k:1. Now, the formula for a point C (x,y,z) to divide line segment joining point A ( ${{x}_{1}},{{y}_{1}},{{z}_{1}}$ ) and B( ${{x}_{2}},{{y}_{2}},{{z}_{2}}$ ) in the ratio m:n is given by-

= $\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\dfrac{m{{z}_{2}}+n{{z}_{1}}}{m+n} \right)$

Here, in this case, we are given that the ratio as k:1. Thus, we will use the above formula with m = k and n = 1. Now, we have,

A ( ${{x}_{1}},{{y}_{1}},{{z}_{1}}$ ) = (1,2,3)

B ( ${{x}_{2}},{{y}_{2}},{{z}_{2}}$ ) = (−3,4,−5)

Now, using the above formula, we can find the coordinate of C (x,y,z) as -

C (x, y, z) = $\left( \dfrac{k{{x}_{2}}+{{x}_{1}}}{k+1},\dfrac{k{{y}_{2}}+{{y}_{1}}}{k+1},\dfrac{k{{z}_{2}}+{{z}_{1}}}{k+1} \right)$

(Since, m = k and n = 1)

Now, putting values of points A ( ${{x}_{1}},{{y}_{1}},{{z}_{1}}$ ) and B( ${{x}_{2}},{{y}_{2}},{{z}_{2}}$ ) as (1, 2, 3) and (-3, 4, -5). Thus, we have,

= $\left( \dfrac{k(-3)+1}{k+1},\dfrac{k(4)+2}{k+1},\dfrac{k(-5)+3}{k+1} \right)$ -- (1)

Since point C (x,y,z) lies on the XY – plane, the z coordinate of C will be zero. Thus, we have,

-5k + 3 = 0

k = $\dfrac{3}{5}$

We can substitute the value of k in (1), we get,

= $\left( \dfrac{-0.8}{1.6},\dfrac{4.4}{1.6},0 \right)$

= (-0.5, 2.75, 0) -- (2)

Thus, the required ratio is k:1, that is 0.6 : 1. Simplifying the ratio, we get,

0.6 : 1 = 3 : 5. Thus, the required ratio is 3 : 5.

We get the positive vector of the point of division from (2). Thus, the vector is given by -0.5i + 2.75j. Here, i denotes unit vector to X-axis and j denotes unit vector to Y-axis.

Note: Generally, while solving the problems related to finding points dividing a line segment in a particular ratio, we always try to use the formula by inserting ratio as k : 1, instead of m:n. This way, we only have to deal with one variable, which makes it easier to deal with the obtained equations.

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