Introduction to Three Dimensional Geometry

A point is on the $x$-axis. What are its $y$-coordinate and $z$-coordinates?

A point is in the XZ-plane. What can you say about its $y$-coordinate?

Name the octants in which the following points lie: $(1,2,3), (4,-2,3), (4,-2,-5), (4,2,-5), (-4,2,-5), (-4,2,5), (-3,-1,6) (-2,-4,-7)$.

Fill in the blanks:

Find the distance between the following pairs of points:

Show that the points $(-2, 3, 5)$, $(1, 2, 3)$ and $(7, 0, -1)$ are collinear.

Verify the following:

Find the equation of the set of points which are equidistant from the points $(1, 2, 3)$ and $(3, 2, -1)$.

Find the equation of the set of points P, the sum of whose distances from A $(4, 0, 0)$ and B $(-4, 0, 0)$ is equal to 10.

Three vertices of a parallelogram $ABCD$ are $A(3,\, -1,\, 2)$, $B(1,\, 2,\, -4)$ and $C(-1,\, 1,\, 2)$. Find the coordinates of the fourth vertex.

Find the lengths of the medians of the triangle with vertices $A(0, 0, 6)$, $B(0, 4, 0)$ and $(6, 0, 0)$.

If the origin is the centroid of the triangle PQR with vertices P(2a, 2, 6), Q(−4, 3b, −10) and R(8, 14, 2c), then find the values of a, b and c.

If $A$ and $B$ be the points $(3, 4, 5)$ and $(−1, 3, −7)$, respectively, find the equation of the set of points $P$ such that $PA^2 + PB^2 = k^2$, where $k$ is a constant.