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Chapter 10: Q 39. (page 846)

Find the distance from the point (2,0,鈭�3) to the plane 3x鈭�4y+5z=1

Short Answer

Expert verified

The distance from the point to the plane is 2

Step by step solution

01

Given information

The planes 3x-4y+5z=1and the point P=(2,0,-3)

02

Calculation

The goal is to calculate the distance between a specified point and the plane. The distance between a point and the plane is calculated as follows:

compNRP鈫�=|N路RP鈫�|鈥�N鈥�

The normal vector of the plane 3x-4y+5z=1is:

N=鉄�3,-4,5鉄�

The point R=0,0,15is on the plane 3x-4y+5z=1

The vector RP鈫�is given by:

RP鈫�=2-0,0-0,-3-15=2,0,-165

03

Calculation

The distance from Pto the plane is:

compNRP鈫�=|N路RP鈫�|鈥�N鈥�=鉄�3,-4,5鉄�路2,0,-165||(3,-4,5)鈭�(Substitution)=|6+0-16|9+16+25(Dot product)=|-10|50=1052(Simplify)=22=2

Therefore, the distance from the point to the plane is 2

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