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Mathematics Question on Vector Algebra

The ratio in which the plane r.(i^\hat i -2j^\hat j + 3k^\hat k ) =17 divides the line joining the points -2i^\hat i+4j^\hat j+7k^\hat k and 3i^\hat i-5j^\hat j+8k^\hat k is

A

5 : 3

B

4 : 5

C

3 : 10

D

10 : 3

Answer

3 : 10

Explanation

Solution

First we need to find the point of intersection between the line and the plane.The direction vector of the line AB is given by the difference between the coordinates of the two points:
d = B - A = ( 3i^\hat i-5j^\hat j+8) - (-2i^\hat i+4j^\hat j+7k^\hat k) = 5i^\hat i - 9j^\hat j + k^\hat k
Now by substituting the parametric form of the line equation into the plane equation:r = A + t.d
Substituting the values: r = (-2i^\hat i+4j^\hat j+7k^\hat k) + t x (5i^\hat i - 9j^\hat j + k^\hat k)
Substituting this into the plane equation:
((-2 + 5t)i^\hat i + (4 - 9t)j^\hat j + (7 + t)k^\hat k) . (i^\hat i - 2j^\hat j + 3k^\hat k) = 17
(-2 + 5t)i^\hat i2 + (4 - 9t)(-2) + (7 + t)3 = 17 (5t - 2) + (-8 + 18t) + (21 + 3t) = 17 26t + 11 = 17 26t = 6 t = 6/26 t = 3/13.
Now, we substitute this value of t back into the parametric equation of the line to find the point of intersection:
r = A + t.d.r = (-2i^\hat i+4j^\hat j+7k^\hat k) +313\frac {3}{13} x (5i^\hat i - 9j^\hat j + k^\hat k) r = (-2i^\hat i+4j^\hat j+7) +\frac {15}{13}$$\hat i - \frac {27}{13}$$\hat j + \frac {3}{13}$$\hat k
r = \frac {15}{13}$$\hat i + (4 - \frac {27}{13})$$\hat j + (7 + \frac {3}{13})$$\hat k
r =\frac {15}{13}$$\hat i + \frac {35}{13}$$\hat j +\frac {100}{13}$$\hat k
Therefore, the point of intersection is:
P = \frac {15}{13}$$\hat i + \frac {35}{13}$$\hat j + \frac {100}{13}$$\hat k
Now, find the ratios by calculating the distances between the points:
AP = (1513−(−2))2+(3513−4)2+(10013−7)2\sqrt{(\frac {15}{13}- (-2))^2 + (\frac {35}{13} - 4)^2 + (\frac {100}{13} - 7)^2}
BP = (3−1513)2+(−5−3513)2+(8−10013)2\sqrt{(3 - \frac {15}{13})^2 + (-5 - \frac {35}{13})^2 + (8 - \frac {100}{13})^2}
Calculating the distances:
AP = 225169+1089169+8100169\sqrt{\frac {225}{169} + \frac {1089}{169} + \frac {8100}{169}} = 10414169\sqrt{\frac {10414}{169}} = 1041413\frac {\sqrt{{10414}}}{13}
BP = 468169+25169+392169\sqrt{\frac {468}{169} +\frac {25}{169} + \frac {392}{169}} = 885169\sqrt{\frac {885}{169}} = 88513\frac {\sqrt{{885}}}{13}
Now express the ratio AP:BP
AP:BP = 1041413\frac {\sqrt{{10414}}}{13} : 88513\frac {\sqrt{{885}}}{13}=10414\sqrt{10414} : 885\sqrt{885}
Simplifying by rationalizing the denominators:
AP:BP = 10414885\frac {\sqrt{10414}} {\sqrt{885}} x 885885\frac {\sqrt{885}} {\sqrt{885}}= 10414×885885×885\frac {\sqrt{10414 \times 885}} {\sqrt{885 \times 885}} =9207690885\frac {\sqrt{9207690}} {{885}}
Therefore, the ratio in which the plane divides the line joining the points is (C) 3:10