Basics of Three Dimensional Coordinate System

$\mathrm{OABCDE}$ is a regular hexagon of side $2 \mathrm{units}$ in the $XY$-plane in the ${\mathrm{I}}^{\mathrm{st}}$ quadrant. $\mathrm{O}$ being the origin and $\mathrm{OA}$ taken along the $X$-axis. A point $\mathrm{P}$ is taken on a line parallel to $Z$-axis through the center of the hexagon at a distance of $3 \mathrm{units}$ from $\mathrm{O}$ in the positive $\mathrm{Z}$ direction. The vector $\overrightarrow{\mathrm{AP}}$ is 

The octant in which the point $\left(2,-4,-7\right)$ lies is

For every point $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ on the $\mathrm{xy}$-plane,

If $\left|x\right|=1, \left|y\right|=2$ and $\left|z\right|=3$, then how many points in $R^3$ are there having co-ordinates $\left(x, y, z\right)?$

Which of the following is a point in third octant?

Which of the following is a point in first octant?

The point $\left(4,5,-6\right)$ lies in

What will the coordinates of the point $C$.If the centroid of a triangle ABC is at the point $(1, 1, 1)$ and the coordinates of $A \text{and} B \text{are} (3, –5, 7) \text{and} (–1, 7, – 6),$ respectively.

What will be the equation for the set of the points $P$ such that the distances from the points $A(3, 4,-5)$ and $B (-2, 1, 4)$ are equal? 

The points$A (1, 2, 3), B (–1, –2, –1), C (2, 3, 2) \text{and} D (4, 7, 6)$ are the vertices of a:

What will the ratio in which the line segment joining the points $(4, 8, 10)$ and $(6, 10, – 8)$ is divided by the $YZ$-plane.

What will be the coordinates of the centroid of the triangle whose vertices are$(x_1 , y_1 , z_1), (x_2 , y_2 , z_2) \text{and} (x_3 , y_3 , z_3).$

What will be the coordinates of the point which divides the line segment joining the points ($1, –2, 3)$and $(3, 4, –5)$ in the ratio $2 : 3$  externally.

What will be the coordinates of the point which divides the line segment joining the points ($1, –2, 3)$and $(3, 4, –5)$ in the ratio $2 : 3$  internally.

Find the distance between the points $P(1, –3, 4) \text{and }Q (– 4, 1, 2)$.

Which one of the following statements is correct regarding the statement given below?

If $A\left(2, 3, 5\right)$ is the starting point, on moving $4$ units from the point $A$ along $x$-axis we moved to point $B$, from that point we moved $10$ units parallel to positive $y$-axis to reach point $C$ and then move $4$ units upwards perpendicular to $xy$ plane to reach the point $D$.

Which one of the following statements is correct regarding the statement given below?

If $A\left(0, 0, 0\right)$ is the starting point, on moving $4.5$ units from the point $A$ along $y$-axis to reach point $B$, from that point we moved $2.5$ units parallel to positive $z$-axis to reach point $C$ and then move $4$ units along positive $x$-axis to reach the point $D$.

Which one of the following statements is correct regarding the statement given below?

If $A\left(0, 0, 0\right)$ is the starting point, on moving $a$ units from the point $A$ along $z$-axis to reach point $B$, from that point we moved $b$ units parallel to positive $x$-axis to reach point $C$ and then move $c$ units along positive $y$-axis to reach the point $D$.

Which one of the following statements is correct regarding the statement given below?

If $A\left(0, 0, 0\right)$ is the starting point, on moving $2$ units from the point $A$ along $z$-axis to reach point $B$, from which we moved $3$ units parallel to positive $y$-axis to reach point $C$ and then move $4$ units perpendicular to $yz$ plane along the positive $x$-axis to reach the point $D$.

Which one of the following statements is correct regarding the statement given below?

If $A\left(0, 0, 0\right)$ is the starting point, on moving $2$ units from the point $A$ along $x$-axis we moved to point $B$, from that point we moved $3$ units parallel to $y$-axis and perpendicular ot $xz$ plane to reach point $C$ and then move $4$ units perpendicular to $xy$ plane to reach the point $D$.