Locate the following points

(i) $(1,-1,3)$ (ii) $(-1,2,4)$ (iii) $(-2,-4,-7)$ (iv) $(-4,2,-5)$

Name the octant in which each of the following points lies.

(i) $(1,2,3$)

(ii) $(4,-2,3)$

(iii) $(4,-2,-5)$

(iv) $(4,2,-5)$

(v) $(-4,2,5)$

(iv) $(-3,-1,6)$

(vii) $(2,-4,-7)$

(viii) $(-4,2,-5)$.

If $A, B, C$ be the feet of perpendiculars from a point $P$ on the $X, Y$ and $Z$-axes respectively, then find the coordinates of $A, B$ and $C$ in each of the following where the point $P$ is

(i) $\mathrm{A}(3,4,2)$

(ii) $B(-5,3,7)$

(iii) $C(4,-3,-5)$

If $A, B$, and $C$ be the feet of perpendiculars from a point $P$ on the $X Y, Y Z$ and $Z X$-planes respectively, then find the coordinates of $A, B$ and $C$ in each of the following where the point $P$ is

(i) $(3,4,5)$

(ii) $(-5,3,7)$

(iii) $(4,-3,-5)$

How far apart are the points $(2,0,0)$ and $(-3,0,0)$ ?

6 Find the distance from the origin to $(6,6,7)$.

Show that, if $x^2+y^2=1$, then the point $\left(x, y, \sqrt{1-x^2-y^2}\right)$ is at a distance 1 unit form the origin.

Show that the point $A(1,-1,3), B(2,-4,5)$ and $C(5,-13,11)$ are collinear.

Three consecutive vertices of a parallelogram $A B C D$ are $A(6,-2,4)$, $B(2,4,-8)$ and $C(-2,2,4)$. Find the coordinates of the fourth vertex.

Show that the $\triangle A B C$ with vertices $A(0,4,1), B(2,3,-1)$ and $C(4,5,0)$ is right angled.

Find the third vertex of triangle whose centroid is origin and two vertices are $(2,4,6)$ and $(0,-2,5)$.

Find the centroid of a triangle, the mid-point of whose sides are $D(1,2,-3), E(3,0,1)$ and $F(-1,1,-4)$.

The mid-points of the sides of a triangle are $(5,7,11),(0,8,5)$ and $(2,3,-1)$. Find its vertices.

If the vertices of a parallelogram $A B C D$ are $A(1,2,3), B(-1,-2,-1)$ and $C(2,3,2)$, then find the fourth vertex $D$.

Find the coordinate of the points which trisect the line segment joining the points $A(2,1,-3)$ and $B(5,-8,3)$.

If the origin is the centroid of a $\triangle A B C$ having vertices $A(a, 1,3)$, $B(-2, b,-5)$ and $C(4,7, c)$, then find the values of $a, b, c$.

If $A(2,2,-3), B(5,6,9), C(2,7,9)$ be the vertices of a triangle. The internal bisector of the angle $A$ meets $B C$ at the point $D$, then find the coordinates of $D$.

Show that the three points $A(2,3,4), B(1,2,-3)$ and $C(-4,1,-10)$ are collinear and find the ratio in which $C$ divides $A B$.

The mid-point of the sides of a triangle are $(1,5,-1),(0,4,-2)$ and $(2,3,4)$. Find its vertices and also find the centroid of the triangle.

Prove that the points $(0,-1,-7),(2,1,-9)$ and $(6,5,-13)$ are collinear. Find the ratio in which the first point divides the join of the other two.

What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?

Match each item given under the Column I to its correct answer given under Column II.

The distance of point $P(3,4,5)$ from the $Y Z$-plane is

What is the length of foot of perpendicular drawn from the point $P$ $(3,4,5)$ on $Y$-axis?

Distance of the point $(3,4,5)$ from the origin $(0,0,0)$ is

If the distance between the points $(a, 0,1)$ and $(0,1,2)$ is $\sqrt{27}$, ther the value of $a$ is

$X$-axis is the intersection of two planes

Equation of $Y$-axis is considered as

The point $(-2,-3,-4)$ lies in the

A plane is parallel to YZ-plane, so it is perpendicular to

The locus of a point for which $y=0$ and $z=0$, is

The locus of a point for which $x=0$ is

If a parallelopiped is formed by planes drawn through the points $(5,8,10)$ and $(3,6,8)$ parallel to the coordinate planes, then the length of diagonal of the parallelopiped is

$L$ is the foot of the perpendicular drawn from a point $P(3,4,5)$ on the $X Y$-plane. The coordinates of point $L$ are

$L$ is the foot of the perpendicular drawn from a point $(3,4,5)$ on $X$-axis. The coordinates of $L$ are

The three axes $O X, O Y$ and $O Z$ determine ............ .

The three planes determine a rectangular parallelopiped which has ............ of rectangular faces.

The coordinates of a point are the perpendicular distance from the ............ on the respectives axes.

The three coordinate planes divide the space into ............ parts.

If a point $P$ lies in $Y Z$-plane, then the coordinates of a point on $Y Z$-plane is of the form ........... .

The equation of $Y Z$-plane is ............. .

If the point $P$ lies on $Z$-axis, then coordinates of $P$ are of the form ......... .

The equation of $Z$-axis, are ............ .

A line is parallel to $X Y$-plane if all the points on the line have equal ........... .

A line is parallel to $X$-axis, if all the points on the line have equal .............. .

$x=a$ represent a plane parallel to .............. .

The plane parallel to YZ-plane is perpendicular to .................... .

The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensions are 10, 13 and 8 units are ............. .

If the distance between the points $(a, 2,1)$ and $(1,-1,1)$ is 5 , then $a$ ......... .

49 If the mid-points of the sides of a triangle $A B, B C$ and $C A$ are $D(1,2,-3)$, $E(3,0,1)$ and $F(-1,1,-4)$, then the centroid of the $\triangle A B C$ is ............... .