2.2 Vectors in three dimensions  (Page 7/14)

Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point $A(2,3,5)$ is the opposite vertex to the origin, then find

1. the coordinates of the other six vertices of the box and
2. the length of the diagonal of the box determined by the vertices $O$ and $A.$

Find the coordinates of point $P$ and determine its distance to the origin.

$\left(y-5\right)\left(z-6\right)=0$

$\left(z-2\right)\left(z-5\right)=0$

${\left(y-1\right)}^2+{(z-1)}^2=1$

${\left(x-2\right)}^2+{(z-5)}^2=4$

Write the equation of the plane passing through point $(1,1,1)$ that is parallel to the xy -plane.

Write the equation of the plane passing through point $(1,\mathrm{-3},2)$ that is parallel to the xz -plane.

Find an equation of the plane passing through points $(1,\mathrm{-3},\mathrm{-2}),$ $(0,3,\mathrm{-2}),$ and $(1,0,\mathrm{-2}).$

Find an equation of the plane passing through points $(1,9,2),$ $(1,3,6),$ and $(1,\mathrm{-7},8).$

Center $C\left(\mathrm{-1},7,4\right)$ and radius $4$

Center $C\left(\mathrm{-4},7,2\right)$ and radius $6$

Diameter $PQ,$ where $P\left(\mathrm{-1},5,7\right)$ and $Q\left(\mathrm{-5},2,9\right)$

Diameter $PQ,$ where $P\left(\mathrm{-16},\mathrm{-3},9\right)$ and $Q\left(\mathrm{-2},3,5\right)$

$P(1,2,3)$ $x^2+y^2+z^2-4z+3=0$

$x^2+y^2+z^2-6x+8y-10z+25=0$

$P\left(3,0,2\right)$ and $Q\left(\mathrm{-1},\mathrm{-1},4\right)$

$P\left(0,10,5\right)$ and $Q\left(1,1,\mathrm{-3}\right)$

$P\left(\mathrm{-2},5,\mathrm{-8}\right)$ and $M\left(1,\mathrm{-7},4\right),$ where $M$ is the midpoint of the line segment $PQ$

$Q\left(0,7,\mathrm{-6}\right)$ and $M\left(\mathrm{-1},3,2\right),$ where $M$ is the midpoint of the line segment $PQ$

Find terminal point $Q$ of vector $\overrightarrow{PQ}=⟨7,\mathrm{-1},3⟩$ with the initial point at $P\left(\mathrm{-2},3,5\right).$

Find initial point $P$ of vector $\overrightarrow{PQ}=⟨\mathrm{-9},1,2⟩$ with the terminal point at $Q\left(10,0,\mathrm{-1}\right).$

$\text{a}=⟨\mathrm{-1},\mathrm{-2},4⟩,$ $\text{b}=⟨\mathrm{-5},6,\mathrm{-7}⟩$

$\text{a}=⟨3,\mathrm{-2},4⟩,$ $\text{b}=⟨\mathrm{-5},6,\mathrm{-9}⟩$

$\text{a}=\text{−}\text{k},$ $\text{b}=\text{−}\mathbf{\text{i}}$

$\text{a}=\text{i}+\text{j}+\text{k},$ $\text{b}=2\text{i}-3\mathbf{\text{j}}+2\text{k}$

$\text{u}=2\text{i}+3\text{j}+4\text{k},$ $\text{v}=\text{−}\text{i}+5\text{j}-\text{k}$

$\text{u}=\text{i}+\text{j},$ $\text{v}=\text{j}-\text{k}$

$\text{u}=⟨2\text{cos}t,\mathrm{-2}\text{sin}t,3⟩,$ $\text{v}=⟨0,0,3⟩,$ where $t$ is a real number.

$\text{u}=⟨0,1,\text{sinh}t⟩,$ $\text{v}=⟨1,1,0⟩,$ where $t$ is a real number.

$\text{a}=3\text{i}-4\text{j}$

$\text{a}=⟨4,\mathrm{-3},6⟩$

$\text{a}=\overrightarrow{PQ},$ where $P\left(\mathrm{-2},3,1\right)$ and $Q\left(0,\mathrm{-4},4\right)$

$\text{a}=\overrightarrow{OP},$ where $P\left(\mathrm{-1},\mathrm{-1},1\right)$

$\text{a}=\text{u}-\text{v}+\mathbf{\text{w}},$ where $\text{u}=\text{i}-\text{j}-\text{k},$ $\text{v}=2\text{i}-\text{j}+\text{k},$ and $\text{w}=\text{−}\text{i}+\text{j}+3\text{k}$