Question

which of the points P(6,2,3),Q(-5,-1,4), and R(0,3,8) is closest to the xz-plane? Which point lies in yz-plane?

Answer 1

Answer:

- Q is closest to the xz-plane

- R belongs to the yz-plane

Step-by-step explanation:

since the xz plane represents all the points (x,0,z), then for a point (x,y,z) the distance is the absolute value of the y coordinate (see note below)

- distance from P to xz plane = P(6,2,3) = |2|=2

- distance from Q to xz plane =Q(-5,-1,4) = |-1|=1

- distance from R to xz plane = R(0,3,8) = |3|=3

then Q is closest to the xz-plane

since the yz-plane represents all the points (0,y,z) then the only point that can satisfy this condition (x=0) is R(0,3,8). Thus R belongs to the yz-plane .

Note:

if we make a vector v from the point to the (x,y,z) to the xz plane (x,0,z) :

v=(x,y,z)-(x,0,z) = (0,y,0)

then the distance will be the modulus of this vector

distance= |v|=√(0²+y²+0²)= √(y²) = |y|