Tag: introduction to 3d geometry

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Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

The coordinate of any point, which lies in $xy$ plane , is

  1. $(x,0,y)$

  2. $(x,x,0)$

  3. $(x, 0, x)$

  4. $(y,0,x)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given that the point lies in $xy$ plane

In $xy$ plane , the coordinate of $z$ will be zero
So $(x,x,0)$ represents a point which lies in $xy$ plane
Therefore option $B$ is correct

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are

  1. three mutually parallel lines

  2. three mutually perpendicular lines

  3. two mutually perpendicular lines and any two parallel

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In three dimensions, the coordinate axes, i.e. x, y and z axes of a rectangular cartesian coordinate system are three mutually perpendicular lines.

The word rectangular is used to indicate perpendicularity among the axes.

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

Who gave the systematic development of analytical geometry for the first time?

  1. Leonard Euler

  2. J. Bernoulli

  3. Rene' Descartes

  4. Pierre Fermat
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

J. bernoulli in a letter of $1715 A.D$ to Leibnitz introduced the three coordinate planes which we are using today

He is the fist one to give the systematic development of analytical geometry
Therefore the correct option is $B$

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

A cube of side 5 has one vertex at the point (1,0,-1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive  z-axis. Find the coordinates of the other vertices of the cube.

  1. <span>&nbsp;</span>(1,0,1)<span>,&nbsp;</span>

  2. <span>&nbsp;</span>(0,-1,0)<span>,&nbsp;</span>

  3. <span>&nbsp;</span>(0,0,-1)<span>,&nbsp;</span>

  4. (1,0,0)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Consider the problem 

Below, are four complete cube face on $XZ-plane,\,(y=0)$  

Given point
$(1,0,-1)$

End of the edge parallel to negative $x-axis$ 
$(0,0-1)$

Origin 
$(0,0,0)$

End of the edge parallel to positive $z-axis $
$(1,0,0)$

And, below 
Four point complete the opposite face of cube. 

consider $P$, end of edge parallel to negative  $y-axis $
$(1,-1,-1)$

Edge from $P$ parallel to positive $z-axis $
$(0,-1,0)$

Edge from $P$ parallel to negative $x-axis $
$(0,-1,-1)$

And 
$(0,-1,0)$

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

The graph of the equation $y^{2}+z^{2}=0$ in three dimensional space is

  1. x- axis

  2. y- axis

  3. z- axis

  4. yz-plane
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Consider the problem 


${y^2} + {z^2} = 0$

$x=0$ and $z=0$

Therefore, 
The graph of the equation 

${y^2} + {z^2} = 0$ is $x-axis$.

Hence, the correct option is $x-axis$.

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

The points $(3,\ 2,\ 0),\ (5,\ 3,\ 2)$ and $(-9,\ 6,\ -3)$, are the vertices of a triangle $ABC.AD$ is the internal bisector of $\angle\ BAC$ which meets $BC$ at $D$. Then the co-ordinates of $D$, are

  1. $\left[ {\dfrac{{17}}{{16}},\ \dfrac{{57}}{{16}},\ \dfrac{{19}}{8}} \right]$

  2. $\left[ {\dfrac{{19}}{{8}},\ \dfrac{{57}}{{16}},\ \dfrac{{17}}{16}} \right]$

  3. $\left[0,\ 0,\ {\dfrac{{17}}{{16}}}\right]$

  4. $\left[{\dfrac{{17}}{{16}}},\ 0,\ 0\right]$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$AD$ is bisector of $ \angle BAC $ 

$ \Rightarrow $ Ratio at $D$ is $ c:b$ where 

$ c = AB = \sqrt{(3-5)^{2}+(2-3)+(0-2)^{2}} $

$ = \sqrt{4+1+4} = \sqrt{9} = 3 $

$ b = AC = \sqrt{(3+9)^{2}+(2-6)^{2}+(0+3)^{2}} $

$ = \sqrt{144+16+9} = \sqrt{162} = 13 $

for point $D$ 

$ x = \dfrac{c(-9)+b(6)}{c+b} = \dfrac{3(-9)+13(5)}{3+13} = \dfrac{38}{16} = \dfrac{19}{8} $

$ y = \dfrac{c(6)+b(3)}{c+b} = \dfrac{3(6)+13(3)}{3+13} = \dfrac{57}{16} $

$ z = \dfrac{c(-3)+b(2)}{c+b} = \dfrac{3(-3)+13(2)}{3+13} = \dfrac{17}{16} $

Hence, point $D$ is $ [\dfrac{19}{8},\dfrac{57}{16},\dfrac{17}{6}] $ 
Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

The points (-5,12), (-2,-3),(9,-10),(6,5) taken in order, form

  1. Parallelogram

  2. rectangle

  3. rhombus

  4. square
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given points $A(-5, 12)\quad B(-2, -3), C(9, -10), D(6, 5)$


Distance $AB=\sqrt{(-5+2)^2+(12+3)^2}=\sqrt{9+225}=\sqrt{234}$
Distance $BC=\sqrt{(-2-9)^2+(-3+10)^2}=\sqrt{121+49}=\sqrt{170}$

Distance $CD=\sqrt{(9-6)^2+(-10-5)^2}=\sqrt{9+225}=\sqrt{234}$
Distance $AD=\sqrt{(-5-6)^2+(12-5)^2}=\sqrt{121+49}=\sqrt{170}$

Distance $AC=\sqrt{(-5-9)^2+(12+10)^2}=\sqrt{196+484}=\sqrt{680}$
Distance $BD=\sqrt{(-2-6)^2+(-3-5)^2}=\sqrt{64+64}=\sqrt{128}$

These points forms a parallelogram, opposite pair of sides are equal and adjacent sides do not form right angles.