\displaystyle \textbf{Question 1: } \text{Name the octants in which the following points lie:}
\displaystyle \text{(i) }(5,2,3)\qquad\text{(ii) }(-5,4,3)\qquad\text{(iii) }(4,-3,5)\qquad\text{(iv) }(7,4,-3)
\displaystyle \text{(v) }(-5,-4,7)\qquad\text{(vi) }(-5,-3,-2)\qquad\text{(vii) }(2,-5,-7)\qquad\text{(viii) }(-7,2,-5)
\displaystyle \text{Answer:}

\displaystyle \text{(i) The x-coordinate, the y-coordinate and the z-coordinate of the point }(5,2,3)
\displaystyle \text{ are all positive. Therefore, the point lies in }XOYZ\text{ octant.}

\displaystyle \text{(ii) The x-coordinate, the y-coordinate and the z-coordinate of the point }(-5,4,3)
\displaystyle \text{ are negative, positive and positive respectively. Therefore, the point lies in }X'OYZ\text{ octant.}

\displaystyle \text{(iii) The x-coordinate, the y-coordinate and the z-coordinate of the point }(4,-3,5)
\displaystyle \text{ are positive, negative and positive respectively. Therefore, the point lies in }XOY'Z\text{ octant.}

\displaystyle \text{(iv) The x-coordinate, the y-coordinate and the z-coordinate of the point }(7,4,-3)
\displaystyle \text{ are positive, positive and negative respectively. Therefore, the point lies in }XOYZ'\text{ octant.}

\displaystyle \text{(v) The x-coordinate, the y-coordinate and the z-coordinate of the point }(-5,-4,7)
\displaystyle \text{ are negative, negative and positive respectively. Therefore, the point lies in }X'OY'Z\text{ octant.}

\displaystyle \text{(vi) The x-coordinate, the y-coordinate and the z-coordinate of the point }(-5,-3,-2)
\displaystyle \text{ are all negative. Therefore, the point lies in }X'OY'Z'\text{ octant.}

\displaystyle \text{(vii) The x-coordinate, the y-coordinate and the z-coordinate of the point }(2,-5,-7)
\displaystyle \text{ are positive, negative and negative respectively. Therefore, the point lies in }XOY'Z'\text{ octant.}

\displaystyle \text{(viii) The x-coordinate, the y-coordinate and the z-coordinate of the point }(-7,2,-5)
\displaystyle \text{ are negative, positive and negative respectively. Therefore, the point lies in }X'OYZ'\text{ octant.}
\displaystyle \\

\displaystyle \textbf{Question 2: } \text{Find the image of:}
\displaystyle \text{(i) }(-2,3,4)\text{ in the YZ-plane}\qquad\text{(ii) }(-5,4,-3)\text{ in the XZ-plane}
\displaystyle \text{(iii) }(5,2,-7)\text{ in the XY-plane}\qquad\text{(iv) }(-5,0,3)\text{ in the XZ-plane}
\displaystyle \text{(v) }(-4,0,0)\text{ in the XY-plane}
\displaystyle \text{Answer:}

\displaystyle \text{(i) Reflection in the YZ-plane changes the sign of the x-coordinate.}
\displaystyle \therefore \text{The image of }(-2,3,4)\text{ is }(2,3,4).

\displaystyle \text{(ii) Reflection in the XZ-plane changes the sign of the y-coordinate.}
\displaystyle \therefore \text{The image of }(-5,4,-3)\text{ is }(-5,-4,-3).

\displaystyle \text{(iii) Reflection in the XY-plane changes the sign of the z-coordinate.}
\displaystyle \therefore \text{The image of }(5,2,-7)\text{ is }(5,2,7).

\displaystyle \text{(iv) Reflection in the XZ-plane changes the sign of the y-coordinate.}
\displaystyle \text{Since }y=0,\text{ the image remains }(-5,0,3).

\displaystyle \text{(v) Reflection in the XY-plane changes the sign of the z-coordinate.}
\displaystyle \text{Since }z=0,\text{ the image remains }(-4,0,0).
\displaystyle \\

\displaystyle \textbf{Question 3: } \text{A cube of side }5\text{ has one vertex at the point }(1,0,-1),\text{ and the three}
\displaystyle \text{edges from this vertex are respectively parallel to the negative }x\text{-axis, negative }y\text{-axis}
\displaystyle \text{and positive }z\text{-axis. Find the coordinates of the other vertices of the cube.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the given vertex be }A(1,0,-1).
\displaystyle \text{Let }AB,\ AD\text{ and }AE\text{ be parallel to the negative }x\text{-axis, negative }y\text{-axis}
\displaystyle \text{and positive }z\text{-axis respectively.}
\displaystyle \text{Since each edge of the cube is }5\text{ units long,}
\displaystyle B=(1-5,0,-1)=(-4,0,-1).
\displaystyle D=(1,0-5,-1)=(1,-5,-1).
\displaystyle E=(1,0,-1+5)=(1,0,4).
\displaystyle \text{The vertex }C\text{ is obtained by moving }5\text{ units along the negative }y\text{-axis from }B.
\displaystyle \therefore C=(-4,-5,-1).
\displaystyle \text{The vertex }F\text{ is obtained by moving }5\text{ units along the positive }z\text{-axis from }B.
\displaystyle \therefore F=(-4,0,4).
\displaystyle \text{The vertex }H\text{ is obtained by moving }5\text{ units along the positive }z\text{-axis from }D.
\displaystyle \therefore H=(1,-5,4).
\displaystyle \text{The vertex }G\text{ is obtained by moving }5\text{ units along the positive }z\text{-axis from }C.
\displaystyle \therefore G=(-4,-5,4).
\displaystyle \therefore \text{The other vertices are }(-4,0,-1),\ (1,-5,-1),\ (1,0,4),\ (-4,-5,-1),
\displaystyle (-4,0,4),\ (1,-5,4)\text{ and }(-4,-5,4).
\displaystyle \\

\displaystyle \textbf{Question 4: } \text{Planes are drawn parallel to the coordinate planes through the points }(3,0,-1)
\displaystyle \text{and }(-2,5,4).\text{ Find the lengths of the edges of the parallelepiped so formed.}
\displaystyle \text{Answer:}
\displaystyle \text{Let }P(3,0,-1)\text{ and }Q(-2,5,4).
\displaystyle \text{The lengths of the edges parallel to the }x\text{-axis, }y\text{-axis and }z\text{-axis are}
\displaystyle |x_2-x_1|,\quad |y_2-y_1|\quad\text{and}\quad |z_2-z_1|\text{ respectively.}
\displaystyle \text{Length of the edge parallel to the }x\text{-axis}=|-2-3|=|-5|=5.
\displaystyle \text{Length of the edge parallel to the }y\text{-axis}=|5-0|=5.
\displaystyle \text{Length of the edge parallel to the }z\text{-axis}=|4-(-1)|=5.
\displaystyle \therefore \text{The lengths of the edges of the parallelepiped are }5,\ 5\text{ and }5\text{ units.}
\displaystyle \\

\displaystyle \textbf{Question 5: } \text{Planes are drawn through the points }(5,0,2)\text{ and }(3,-2,5)\text{ parallel}
\displaystyle \text{to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.}
\displaystyle \text{Answer:}
\displaystyle \text{Let }P(5,0,2)\text{ and }Q(3,-2,5).
\displaystyle \text{The lengths of the edges parallel to the }x\text{-axis, }y\text{-axis and }z\text{-axis are}
\displaystyle |x_2-x_1|,\quad |y_2-y_1|\quad\text{and}\quad |z_2-z_1|\text{ respectively.}
\displaystyle \text{Length of the edge parallel to the }x\text{-axis}=|3-5|=|-2|=2.
\displaystyle \text{Length of the edge parallel to the }y\text{-axis}=|-2-0|=|-2|=2.
\displaystyle \text{Length of the edge parallel to the }z\text{-axis}=|5-2|=3.
\displaystyle \therefore \text{The lengths of the edges of the rectangular parallelepiped are }2,\ 2\text{ and }3\text{ units.}
\displaystyle \\

\displaystyle \textbf{Question 6: } \text{Find the distances of the point }P(-4,3,5)\text{ from the coordinate axes.}
\displaystyle \text{Answer:}
\displaystyle \text{Given point }P(-4,3,5).
\displaystyle \text{Distance of }P\text{ from the }x\text{-axis}=\sqrt{y^2+z^2}=\sqrt{3^2+5^2}=\sqrt{9+25}=\sqrt{34}.
\displaystyle \text{Distance of }P\text{ from the }y\text{-axis}=\sqrt{x^2+z^2}=\sqrt{(-4)^2+5^2}=\sqrt{16+25}=\sqrt{41}.
\displaystyle \text{Distance of }P\text{ from the }z\text{-axis}=\sqrt{x^2+y^2}=\sqrt{(-4)^2+3^2}=\sqrt{16+9}=\sqrt{25}=5.
\displaystyle \therefore \text{The required distances are }\sqrt{34},\ \sqrt{41}\text{ and }5\text{ units respectively.}
\displaystyle \\

\displaystyle \textbf{Question 7: } \text{The coordinates of a point are }(3,-2,5).\text{ Write down the coordinates of}
\displaystyle \text{seven points such that the absolute values of their coordinates are the same as those}
\displaystyle \text{of the coordinates of the given point.}
\displaystyle \text{Answer:}
\displaystyle \text{The given point is }A(3,-2,5).
\displaystyle \text{The absolute values of its coordinates are }3,\ 2\text{ and }5.
\displaystyle \text{Hence the remaining seven points are obtained by changing the signs of one or more coordinates.}
\displaystyle B(3,2,5),\quad C(-3,-2,5),\quad D(3,-2,-5),
\displaystyle E(-3,2,5),\quad F(3,2,-5),\quad G(-3,-2,-5),\quad H(-3,2,-5).
\displaystyle \\

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