Section 1
Cartesian Coordinates; Distance between Points
In Exercises 1-8, Sketch the graph of the point in space which has the given coordinates.$(1,2,1)$
Sketch the graph of the point in space which has the given coordinates. $(3,2,1)$
Sketch the graph of the point in space which has the given coordinates.$(0,2,3)$
Sketch the graph of the point in space which has the given coordinates. $(4,0,2)$
Sketch the graph of the point in space which has the given coordinates.$(4,3,0)$
Sketch the graph of the point in space which has the given coordinates. $(0,0,5)$
Sketch the graph of the point in space which has the given coordinates. $(3,0,0)$
Sketch the graph of the point in space which has the given coordinates.$(0,4,0)$
In Exercises 9-14, State the condition(s) that must be satisfied by the coordinates of every point ( $x, y, z$ ) lying in the given plane(s). $x y$-plane
State the condition(s) that must be satisfied by the coordinates of every point ( $x, y, z$ ) lying in the given plane(s). $y z$-plane
State the condition(s) that must be satisfied by the coordinates of every point ( $x, y, z$ ) lying in the given plane(s).$x z$-plane
State the condition(s) that must be satisfied by the coordinates of every point ( $x, y, z$ ) lying in the given plane(s). $x y$-and $y z$-planes
State the condition(s) that must be satisfied by the coordinates of every point ( $x, y, z$ ) lying in the given plane(s).$x y$ - and $x z$-planes
State the condition(s) that must be satisfied by the coordinates of every point ( $x, y, z$ ) lying in the given plane(s). $y z$ - and $x z$-planes
In Exercises 15-20, Find values for $x, y$, and $z$ so that the two ordered triples are equal. $(3, y, z+1),(3,4,5)$
Find values for $x, y$, and $z$ so that the two ordered triples are equal. $(x, 5, z-2),(4,5,2)$
Find values for $x, y$, and $z$ so that the two ordered triples are equal.$(x+1, y-3, z+4),(-2,3,6)$
Find values for $x, y$, and $z$ so that the two ordered triples are equal.$(x-2, y+2, z-4),(0,0,0)$
Find values for $x, y$, and $z$ so that the two ordered triples are equal. $(x+4,3, z+5),(4, y+3,2)$
Find values for $x, y$, and $z$ so that the two ordered triples are equal. $(1, y-3, z+2),(x-5,4,6)$
In Exercises 21-28, Find the distance between the points S and T. $\mathbf{S}(1,1,2), \mathbf{T}(2,3,4)$
Find the distance between the points S and T. $\mathbf{S}(-1,1,3), \mathbf{T}(0,-1,1)$
Find the distance between the points S and T. $\mathbf{S}(2,-1,5), \mathbf{T}(0,2,-1)$
Find the distance between the points S and T.$\mathbf{S}(3,1,-3), \mathbf{T}(5,4,3)$
Find the distance between the points S and T. $\mathbf{S}(2,1,0), \mathbf{T}(3,5,8)$
Find the distance between the points S and T.$\mathbf{S}(-2,1,-3), \mathbf{T}(2,2,5)$
Find the distance between the points S and T.$\mathbf{S}(4,5,7), \mathbf{T}(6,6,5)$
Find the distance between the points S and T. $\mathbf{S}(5,3,-6), \mathrm{T}(2,-1,6)$
Use the distance formula and the converse of the Pythagorean theorem to show that the points $\mathbf{S}(3,5,2), \mathbf{T}(2,3,-1)$, and $\mathbf{U}(6,1,-1)$ are the vertices of a right triangle.
Show that the points $\mathbf{S}(6,3,4), \mathbf{T}(2,1,-2)$, and $\mathbf{U}(4,-1,10)$ are the vertices of an isosceles triangle.
Show that the points $\mathbf{S}(2,-1,3), \mathbf{T}(4,2,1)$, and $\mathbf{U}(-2,-7,7)$ lie on the same line.
Show that the midpoint of the segment with endpoints $\mathbf{S}_1\left(x_1, y_1, z_1\right)$ and $\mathbf{S}_2\left(x_2, y_2, z_2\right)$ is the point $\mathbf{S}_{\mathrm{m}}\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$.
Find an equation whose graph is the set of all points located a distance of 3 units from $\mathbf{S}(2,3,4)$.
Find an equation whose graph is the set of all points equidistant from $\mathbf{S}(-1,4,0)$ and $\mathbf{T}(2,-1,1)$.
Find an equation whose graph is the set of all points the sum of whose distances from $\mathbf{S}(0,4,0)$ and $\mathbf{T}(0,-4,0)$ is 10 units.
Show that an equation of the sphere with radius 3 and center $\mathbf{S}(1,-2,2)$ is $x^2+y^2+z^2-2 x+4 y-4 z=0$.
Show that an equation of the sphere with radius $r$ and center $\mathbf{S}\left(x_1, y_1, z_1\right)$ is $\left(x-x_1\right)^2+\left(y-y_1\right)^2+\left(z-z_1\right)^2=r^2$.
By completing the squares in $x, y$, and $z$, find the radius and the coordinates of the center of the sphere with equation
$$x^2+y^2+z^2-2 z+8 y-8=0$$