 Questions & Answers    Question Answers

# Find the ratio in which the XY - plane divides AB if coordinates of A is (1,2,3) and B is (-3,4,-5). Also find the positive vector of the point of division.  Answer Verified
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.

Bookmark added to your notes.
View Notes
How to Find The Median?  Equation Of Plane In Normal Form Easy Method  Argand Plane  Cartesian Plane  Equation Plane  Motion in a Plane  CBSE Class 6 Maths Chapter 12 - Ratio and Proportion Formulas  Intercept Form of the Equation of a Plane  Determinant to Find the Area of a Triangle  Visualising Circular Motion in Vertical Plane  