Question

The position vectors of points A and B are \[i - j + 3k\] and \[3i + 3j + 3k\;\]respectively. The equation of a plane is\[r.\left( {5i + 2j - 7k} \right) + 9 = 0\]. The points A and B
A) Lie on the plane
B) Are on the same side of the plane
C) Are on the opposite side of the plane
D) None of these

Answer 1

Hint: in this question, we have to find about the position two given point on a plane. In order to find this, we simply compare the equation of given plane with standard equation of plane and then by putting the value of all parameter of plane and position vector of points we get relative position of point.



Formula Used:Equation of required plane is given by
\[\overrightarrow r .\overrightarrow n + d = 0\]
This is a vector form of plane
Where
\[\overrightarrow r \]Is a position vector of any arbitrary point.
\[\overrightarrow n \]normal vector to the plane .
d is some constant



Complete step by step solution:Equation of required plane is given by
\[\overrightarrow r .\overrightarrow n + d = 0\]
\[\overrightarrow r \]Is a position vector of any arbitrary point.
\[\overrightarrow n \]normal vector to the plane .
d is some constant
In this question equation of a plane is given as \[r.\left( {5i + 2j - 7k} \right) + 9 = 0\]
It is in the form of \[\overrightarrow r .\overrightarrow n + d = 0\]
After comparison we get
\[\overrightarrow n = 5i + 2j - 7k\]and \[d = 9\]
Position vector of A\[ = i - j + 3k\]
Position vector of B \[ = 3i + 3j + 3k\;\]
For point A
Put \[a = A = i - j + 3k\], \[\overrightarrow n = 5i + 2j - 7k\], \[d = 9\]in equation \[\overrightarrow r .\overrightarrow n + d = 0\]
\[(i - j + 3k).(5i + 2j - 7k) + 9\]
\[5 - 2 - 21 + 9 < 0\]
For point B
Put\[a = B = 3i + 3j + 3k\;\], \[\overrightarrow n = 5i + 2j - 7k\], \[d = 9\]in equation \[\overrightarrow r .\overrightarrow n + d = 0\]
\[(3i + 3j + 3k).(5i + 2j - 7k) + 9\]
\[15 + 6 - 21 + 9 > 0\]
Now it is clear that point A and point B are on opposite side of a plane



Option ‘C’ is correct



Note: Concept of the solution is the same as we assume in the number line i.e if one number is negative and another number is positive then we say that both numbers are present at opposite sides of origin.
Similarly, if the position of one point is negative and another point position is positive then we can say that both points lie on opposite sides of origin. In this case we take the plane as an origin.