Distance from a Point to a Plane

In the distance formula, we usually find the distance between two points on a plane. But in this topic, we will discuss the distance from a point to a plane. The distance formula tends to be used when you know the coordinates of the points. We need to substitute the formula to find the two points on a given plane. To locate a given point on a plane surface, you need a coordinate axis. The centre is the x-axis and the point taken from a distance is  the y-axis.

Distance from a point to a plane

• The shortest distance between a point and a plane refers to the perpendicular length drawn from the point onto the plane. 

Distance between line and plane formula: 

• The distance of a point from a plane can be calculated in vector form and cartesian form. 

• Vector form – Let us consider a point B whose position vector is a→ and on Plane 1. The equation of the plane is r. N =D. (Vector N is normal to the plane). 

Now, let’s consider another Plane 2 through point B parallel to Plane 1. The equation of the Plane 2 is (r –a ). N =0. This can also be written as r .N =a .N. 

Therefore, the distance of plane two from its origin is (a. n→ )/|n|. Thus, the distance from point B to Plane 1 is d =  a .n→/|n| .

• Cartesian Form – let us assume that point P (x1, y1, z1) has a position vector A. Let the cartesian equation of the given plane by Ax + By +Cz = D. 

Then, a = x1 i + y1 j+ z1 k

And, N = A i + B j + C k 

In one of the earlier points, we have established that the distance of the point from the plane is the perpendicular drawn from the projection onto the plane. 

Therefore,( x1i + y1 j + z1 k) . (A i + B j + C k)/(A2+B2+C2)

= Ax1+By1 +Cz1(A2+B2+C2)

Distance from a point to a plane solved examples. 

Example 1

Find the distance between the Plane 6 i-3 j+2 k=4 and Point (2, 5, -3). 

a=2 i+5 j-3k , N=6 i-3 j+2k, and d = 4. 

d  = Ax1+By1 +Cz1(A2+B2+C2)

Therefore, (2 i+5 j-3k).(6 i-3 j+2k)=-9-4=-13

36+9+4 = 49

Therefore, the distance between point and plane is 13/7.  

Example 2

Find the distance of a plane 3x – 4y + 12z = 3 from the point of its origin. 

A = 3, B = -4, C = 12, and D = 3

Since we are calculating the distance of a plane from the point of origin, the point is (0, 0, 0). 

Distance = d = (x1 i + y1 j + z1 k) . (A i + B j + C k)(A2+B2+C2) = Ax1+By1+Cz1/(A2+B2+C2)

 d = 3 ×0 +-4 ×0+12 ×0 -3=-3

A2 + B2 + C2=13

Therefore, the distance of point from plane = 3/13

Example 3

Find the distance between a point (3, -2, 1) and plane 2x – y + 2z + 3 = 0 

A = 2, B = -1, C = 2, and D = -3

Distance = d = (x1 i + y1 j + z1 k) ×( A i + B j + C k)-D/(A2+B2+C2)

d = 2 ×3 +-1 ×-2+2 ×1+ 3

d = (6 +2+2+ 3)/(9

d = 13/3

Therefore, the distance between a point (3, -2, 1) and plane 2x – y + 2z + 3 = 0 is 13/3 . 

Example 4 

Find the distance between a point (2, 3, -5) and plane x + 2y – 2z = 9

A = 1, B = 2, C = -2, and D = 9

Distance = d = (x1 i + y1 j + z1 k) . (A i + B j + C k)(A2+B2+C2)= Ax1+By1+Cz1/(A2+B2+C2)

d = 1 ×2 +2 ×3 +-2 ×-5- 9=9

(A2+B2+C2)=3

Therefore, the distance between a point (2, 3, -5) and plane x + 2y – 2z = 9 is 3. 

Example 5

Find the distance between a point (-6, 0, 0) and plane 2x – 3y + 6z – 2 = 0

A = 2, B = -3, C = 6, and D = 2

Distance = d  = (x1 i + y1 j + z1 k) . (A i + B j + C k)-D(A2+B2+C2) = Ax1+By1+Cz1/(A2+B2+C2) 

d = 2 ×-6 +-3 ×0+6×0- 2=-14

(A2+B2+C2) =7 Hence, distance =2 units.

Other distance equations in three-dimension geometry:

• Distance between parallel lines 

The formula to determine the distance between two parallel lines is – 

d = PT = b ×(a2– a1)/|b|

• Distance between two skew lines 

Vector form – The shortest distance between two skew lines can be determined by 

d = |(b1 ×b2) .(a2– a1)|/|b1 × b2|

Conclusion

To find the distance from a point to a plane is ( x1i + y1 j + z1 k) . (A i + B j + C k)/ A2 + B2 + C2 

= (Ax1+By1+Cz1-D)/(A2+B2+C2)

The distance formula tends to be used when you know the coordinates on the points you need to substitute the formula to find the Distance from a point to a plane. To locate a given point on a plane surface, you need a coordinate axis. You can also use this formula as a distance formula calculator.