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Practice each question, then open the answer or worked solution.
The locus of a point for which $x = 0$ is
(a) xy - plane
(b) yz - plane
(c) zx - plane
(d) none of these
Ans: Option (b)
Given: $x = 0$.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
$x = 0$ on the $yz -$ plane.
Hence, the locus is $yz -$ plane.
If a parallelepiped 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 parallelepiped is
(a) \[2\sqrt 3\]
(b) \[3\sqrt 2\]
(c) \[\sqrt 2\]
(d) \[\sqrt 3\]
Ans: Option (a)
Given: $(5,8,10)$ and $(3,6,8)$.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
The length of diagonal is:
$\Rightarrow AB = \sqrt {{{(5 - 3)}^2} + {{(8 - 6)}^2} + {{(10 - 8)}^2}}$
$\Rightarrow AB = \sqrt {4 + 4 + 4}$
$\Rightarrow AB = 2\sqrt 3$
Hence, the length of diagonal of the parallelepiped is $2\sqrt 3 .$
$L$ is the foot of the perpendicular drawn from a point $P(3,4,5)$ on the $xy -$ plane. The coordinates of point $L$ are
(a) (3,0,0)
(b) (0,4,5)
(c) (3,0,5)
(d) none of these
Ans: Option (d)
Given: $P(3,4,5)$.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
On the $xy -$ plane, $z = 0$.
Hence, the coordinates of point $L$ are $(3,4,0)$.
$L$ is the foot of the perpendicular drawn from a point $(3,4,5)$ on $x -$ axis. The coordinates of $L$ are
(a) (3,0,0)
(b) (0,4,0)
(c) (0,0,5)
(d) none of these
Ans: Option (a)
Given: $(3,4,5)$.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
On the $x -$ axis, $y = 0$ and $z = 0$.
Hence, the coordinates of $L$ are $(3,0,0)$.
The three axes $OX,$$OY,$$OZ$ determine.
Ans: Given: $OX,$$OY,$$OZ$
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
The three axes $OX,$$OY,$$OZ$ determine $3$ coordinate planes.
Hence, $OX,$$OY,$$OZ$ determine $3$ coordinate planes.
The three planes determine a rectangular parallelepiped which has $\_\_\_\_\_\_\_\_$of rectangular faces.
Ans: Given: The three planes determine a rectangular parallelepiped.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
The rectangular parallelepiped determined by three planes has three pairs of rectangular faces.
Hence, the rectangular parallelepiped has three pairs of rectangular faces.
The coordinates of a point are the perpendicular distance from the $\_\_\_\_\_\_\_\_$ on the respective axes.
Ans: Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
The coordinates of a point are the perpendicular distance from the given points on the respective axes.
The three coordinate planes divide the space into$\_\_\_\_\_\_\_\_$parts.
Ans: Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
The three coordinate planes divide the space into $8$ parts.
Hence, the space is divided into $8$ parts.
If a point $P$ lies on $yz -$ plane, then the coordinates of a point on $yz -$ plane is of the form __________.
Ans: Given: $P$ lies on $yz -$ plane.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
On the $yz -$ plane, $x = 0$.
Hence, the coordinates are of the form $(0,y,z)$.
The equation of $yz -$ plane is __________.
Ans: Given: $yz -$ plane.
Three dimensional geometry involves the mathematics of shapes in $3D$ space and the three coordinates in $XYZ$ plane.
For any point on the $yz -$ plane, the $x -$ coordinate is $0$.
Hence, the equation of$yz -$ plane is $x = 0$.