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

Given points lie in octant

Octants Coordinates	I	II	III	IV	V	VI	VII	VIII
x	+	–	–	+	+	–	–	+
y	+	+	–	–	+	+	–	–
z	+	+	+	+	–	–	–	–

The $x,y,z$-coordinates of point $(1,2,3)$ are all positive. So, point lies in octant $I$.

The $x,y,z$ -coordinates of point $(4,-2,3)$ are positive, negative, and positive respectively. So, point lies in octant $IV$.

Octants Coordinates	I	II	III	IV	V	VI	VII	VIII
x	+	–	–	+	+	–	–	+
y	+	+	–	–	+	+	–	–
z	+	+	+	+	–	–	–	–

The $x,y,z$-coordinates of point $(4,-2,-5)$ are positive, negative, and negative respectively. So, point lies in octant $VIII$.

The $x,y,z$-coordinates of point $(4,2,-5)$ are positive, positive, and negative respectively. So, point lies in octant $V$

Octants Coordinates	I	II	III	IV	V	VI	VII	VIII
x	+	–	–	+	+	–	–	+
y	+	+	–	–	+	+	–	–
z	+	+	+	+	–	–	–	–

The $x,y,z$-coordinates of point $(-4,2,-5)$ are negative, positive, and negative respectively. So, point lies in octant $VI$

The $x,y,z$-coordinates of point $(-4,2,5)$ are negative, positive, and positive respectively. So, point lies in octant $II$.

Octants Coordinates	I	II	III	IV	V	VI	VII	VIII
x	+	–	–	+	+	–	–	+
y	+	+	–	–	+	+	–	–
z	+	+	+	+	–	–	–	–

The $x,y,z$-coordinates of point $(-3,-1,6)$ are negative, negative, and positive respectively. So, point lies in octant $III$.

The $x,y,z$-coordinates of point $(-2,-4,-7)$ are negative, negative, and negative respectively. So, point lies in octant $VII$.