Question

The co-ordinates of the foot of the perpendicular drawn from the origin to the plane 2x - 3y + 4z = 29 are

• A. (2,3,4)
• B. (2,-3,4)
• C. (2,-3,-4)
• D. (-2,-3,4)

Answer 1

The Correct Option is B

The normal vector of the plane is given by the coefficients of x, y, and z in the equation, which is (2, -3, 4).
This vector is perpendicular to the plane. 
The direction vector of the line perpendicular to the plane is the same as the normal vector of the plane.
 Therefore, the direction vector is (2, -3, 4). 
To find the point of intersection, we can parameterize the line in terms of a parameter
 \(t: x = 2t\ y = -3t\ z = 4t \)
Substituting these values into the equation of the plane, we get: 
\(2(2t) - 3(-3t) + 4(4t)\)
\(=  4t + 9t + 16t \)
\(= 29 t\)
 \(= 1\)
 Therefore, the point of intersection, which is the foot of the perpendicular from the origin to the plane, is: 
\(x = 2(1) = 2 y = -3(1) = -3 z = 4(1) = 4 \)
So, the coordinates of the foot of the perpendicular are\( (2, -3, 4). \)
Therefore, the correct option is (B) (2, -3, 4).