Question

Name the octants in which the following points lie: (1, 2, 3), (4, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5), (–3, –1, 6), (2, –4, –7)

Answer 1

The octant in which a point ( x,y,z ) lies can be found from the sign of its coordinates.

(i)

The point ( 1,2,3 ) , where x=1,y=2,z=3 , has all the coordinates with positive sign.

A point with all positive coordinates lie in octant I.

Therefore the given point ( 1,2,3 ) , lies in octant I.

(ii)

The point ( 4,−2,3 ) , where x=4,y=−2,z=3 , has positive x and z coordinates , while the sign of y coordinate is negative.

A point with positive x and z coordinates and negative y coordinate lies in octant IV.

Therefore, the given point ( 4,−2,3 ) , lies in octant IV.

(iii)

The point ( 4,−2,−5 ) , where x=4,y=−2,z=−5 , has positive x coordinate, while the sign of y and z coordinate is negative.

A point with positive x coordinates and negative y and z coordinate lies in octant VIII.

Therefore, the given point ( 4,−2,−5 ) lies in octant VIII.

(iv)

The point ( 4,2,−5 ) , where x=4,y=2,z=−5 , has positive x and y coordinates , while the sign of z coordinate is negative.

A point with positive x and y coordinates and negative z coordinate lies in octant V.

Therefore, the given point ( 4,2,−5 ) lies in octant V.

(v)

The point ( −4,2,−5 ) , where x=−4,y=2,z=−5 , has positive y coordinate , while the sign of x and z coordinates is negative.

A point with positive y coordinates and negative x and z coordinate lies in octant VI.

Therefore, the given point ( −4,2,−5 ) lies in octant VI.

(vi)

The point ( −4,2,5 ) , where x=−4,y=2,z=5 , has positive y and z coordinates, while the sign of x coordinate is negative.

A point with positive y and z coordinates and negative x coordinate lies in octant II.

Therefore, the given point ( −4,2,5 ) lies in octant II.

(vii)

The point ( −3,−1,6 ) , where x=−3,y=−1,z=6 , has positive z coordinate, while the sign of x and y coordinate s is negative.

A point with positive z coordinate and negative x and y coordinates lies in octant III.

Therefore, the given point ( −3,−1,6 ) lies in octant III.

(viii)

The point ( −2,−4,−7 ) , where x=−2,y=−4,z=−7 , has all the coordinates with negative sign.

A point with all negative coordinates lie in octant VII.

Therefore the given point ( −2,−4,−7 ) , lies in octant VII.