Your cart is empty.
Practice each question, then open the answer or worked solution.
If the point $P$ lies on $z -$ axis, then coordinates of $P$ are of the form _________.
Ans: Given: $P$ lies on $z -$ axis.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
On the $z -$ axis, $x = 0$ and $y = 0$.
Hence, the coordinates are of the form $(0,0,z)$.
The equations of $z -$ axis are __________.
Ans: Given: $z -$ axis
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
On the $z -$ axis, the coordinates of any point are of the form $(0,0,z)$.
Hence, the equation of$z -$ axis is $x = 0$ and $y = 0$.
A line is parallel to $xy -$ plane if all the points on the line have equal _________.
Ans: Given: A line is parallel to $xy -$ plane. Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
A line is parallel to $xy -$ plane if every point on it is at the same distance from the $xy -$ plane.
The distance of any point from $xy -$ plane is $z$. Hence, all the points on the line have equal $z -$ coordinate.
A line is parallel to $x -$ axis if all the points on the line have equal __________.
Ans: Given: A line is parallel to $x -$ axis.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
A line is parallel to $x -$ axis if every point on it maintains a constant distance from$y -$ axis and $z -$ axis.
Hence, all the points on the line have equal $y$ and $z -$ coordinates.
$x = a$ represents a plane parallel to _________.
Ans: Given: $x = a$
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
The locus of a point $P(x,y,z)$ is $x = a$.
Every point $P$ has a constant $x -$ coordinate.
$x$ is the distance of $P$ from the $yz -$ plane. The plane $x = a$ is at constant distance $a$ from the $yz -$ plane.
Hence, $x = a$ represents a plane parallel to the$yz -$ plane.
The plane parallel to $yz -$ plane is perpendicular to __________.
Ans: Given: $yz -$ plane.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
The plane parallel to $yz -$ plane is perpendicular to the $x -$ axis.
Hence, the plane which is parallel to $yz -$ plane is perpendicular to the $x -$ axis.
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 _________.
Ans: Given: Dimensions of rectangular rooms are $10,$$13$ and $8$ units.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
Given dimensions are $a = 10,$$b = 13$ and $c = 8$.
The length of string required$= \sqrt {\left( {{a^2} + {b^2} + {c^2}} \right)}$
Required length $= \sqrt {\left( {{{10}^2} + {{13}^2} + {8^2}} \right)}$
Required length $= \sqrt {\left( {100 + 169 + 64} \right)}$
Required length $= \sqrt {333}$
Hence, the length of the longest piece of a string is $\sqrt {333}$.
If the distance between the points $(a,2,1)$ and $(1, - 1,1)$ is $5$, then $a$ ________.
Ans: Given: The distance between $(a,2,1)$ and $(1, - 1,1)$ is $5$.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
Given points are $(a,2,1)$ and $(1, - 1,1)$.
Given $\sqrt {{{\left( {a - 1} \right)}^2} + {{\left( {2 + 1} \right)}^2} + {{\left( {1 - 1} \right)}^2}} = 5$
$\Rightarrow {\left( {a - 1} \right)^2} + 9 + 0 = 25$
$\Rightarrow {a^2} - 2a - 15 = 0$
$\Rightarrow \left( {a - 5} \right)\left( {a + 3} \right) = 0$
$\Rightarrow a = 5$ or $- 3$
Hence, $a = 5$ or $- 3$.
If the mid-points of the sides of a triangle $AB;$$BC;$$CA;$ are $D(1,2, - 3)$, $E(3,0,1)$ and $F( - 1,1, - 4)$, then the centroid of the triangle $ABC$ is ________.
Ans: Given: The mid-points of the sides of a triangle$AB;$$BC;$$CA;$ are$D(1,2, - 3)$, $E(3,0,1)$ and $F( - 1,1, - 4)$.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
Given that the mid-points of the sides of$\Delta ABC$ are $D(1,2, - 3)$, $E(3,0,1)$ and $F( - 1,1, - 4)$.
The centroid of the triangle formed by the midpoints of the sides of a triangle is the same as the centroid of the triangle.
So, the centroid of $\Delta DEF$ and the centroid of $\Delta ABC$ are the same.
Centroid of $\Delta ABC = G\left( {\dfrac{{1 + 3 - 1}}{3},\dfrac{{2 + 0 + 1}}{3},\dfrac{{ - 3 + 1 - 4}}{3}} \right)$
Centroid of $\Delta ABC = G\left( {1,1, - 2} \right)$
Hence, the centroid of the triangle is $\left( {1,1, - 2} \right)$.
Match each item given under the column ${C_1}$ to its correct answer given under column ${C_2}$.
Column ${C_1}$ | Column ${C_2}$ |
(a) In $xy -$ plane | (i) ${\text{I}}$st octant |
(b) Point $(2,3,4)$ lies in the | (ii) $yz -$ plane |
(c) Locus of the points having $x$ coordinate $0$ is | (iii) $z -$ coordinate is zero |
(d) A line is parallel to $x -$ axis if and only | (iv) $z -$ axis |
(e) If $x = 0$, $y = 0$ taken together will represent the | (v) plane parallel to $xy -$ plane |
(f)$z = c$represent the plane | (vi) if all the points on the line have equal $y$ and $z -$ coordinates. |
(g) Planes $x = a$, $y = b$ represent the line | (vii) from the point on the respective |
(h) Coordinates of a point are the distances from the origin to the feet of perpendiculars | (viii) parallel to $z -$ axis. |
(i) A ball is the solid region in the space enclosed by a | (ix) disc |
(j) Region in the plane enclosed by a circle is known as a | (x) sphere |
Ans: Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
(a) In $xy -$ plane, $z -$ coordinate is zero.
(b) The point $(2,3,4)$ lies in ${\text{I}}$st octant.
(c) Locus of the points having $x -$ coordinate zero is $yz -$ plane.
(d) A line is parallel to $x -$ axis if and only if all the points on the line have equal $y$ and $z -$ coordinates.
(e)$x = 0$, $y = 0$ represent $z -$ axis.
(f)$z = c$ represents the plane parallel to $xy -$ plane.
(g) The plane $x = a$ is parallel to $yz -$ plane.
The plane $y = b$ is parallel to $xz -$ plane.
The planes $x = a$ and $y = b$ is the line of intersection of these planes.
The line of intersection of $yz -$ plane and $xz -$ plane is $z -$ axis.
Thus, the line of intersection of planes $x = a$ and $y = b$ is a line parallel to $z -$ axis.
(h) The coordinates of a point are the distances from the origin to the feet of perpendicular from the point on the respective axis.
(i) A ball is the solid region in the space enclosed by a sphere.
(j) The region in the plane enclosed by a circle is known as a disc.
Hence,${\text{(a) - (iii), (b) - (i), (c) - (ii), (d) - (vi), (e) - (iv), (f) - (v), (g) - (viii), (h) - (vii), (i) - (x), (j) - (ix)}}$.
About NCERT
NCERT Exemplar for Class 11 chapter 12 deals with the Introduction to Three Dimensional Geometry. It involves the vector-algebra approach towards three-dimensional geometry with minimal use of complicated three-dimensional diagrams like triangles, etc. This chapter also explains the standard properties of lines and planes as well as three-dimensional space.
Three Dimensional Geometry includes three coordinates- x, y and z. They are mainly used to find the exact location of a point on a plane. The three coordinates are together known as the three-dimensional space. The three coordinate axes are then paired together- xy, yz, and zx which form three coordinate planes. We also discuss here the coordinates of a point in three-dimensional space as arbitrary point P is marked as (x0y0z0). These space coordinates are called Cartesian coordinates of P or rectangular coordinates of P. Next in line, we talk about the signs of coordinates of a point in which the sign depends on the octant in which the point lies. For example, in the first octant, all coordinates are positive and by the seventh octant, all the coordinates turn negative. In the fourth octant, x and z are positive while y is negative and in the sixth octant, it is the opposite, i.e., x and z are negative and y is positive. There are two formulas included in this chapter:
The Distance Formula between two points (A, B)= √(x2-x1)2+(y2-y1)2+(z2-z1)2 which is also the length of the diagonal.
The Section Formula where the coordinates of point R that divides a line segment joining points P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m:n is calculated as [(mx2+nx1)/(m+n)]+[(my2+ny1)/(m+n)]+[(mz2+nz1)/(m+n)], [(mx2-nx1)/(m-n)]+[(my2-ny1)/(m-n)]+[(mz2-nz1)/(m-n)]