Elements of Vector Algebra and Affine Coordinate Systems

In a triangle $A B C$ the medians $A D, B E$ and $C F$ are drawn. Find the sum of the vectors $\overrightarrow {A D}+\overrightarrow {B E}+\overrightarrow {C F}$.

Points $E$ and $F$ are the midpoints of the sides $A B$ and $C D$ of a quadrilateral $A B C D$. Prove that $\overrightarrow {E F} = (\overrightarrow {B C}+\overrightarrow {A D}) / 2$. Deduce from this the converse theorem on the midline of a trapezoid.

The vectors $\overrightarrow {A C} = \mathbf{a}$ and $\overrightarrow {B D} = \mathbf{b}$ are the diagonals of a parallelogram $A B C D$. Express in terms of the vectors $\mathbf{a}$ and $\mathbf{b}$ the vectors $\overrightarrow {A B}, \overrightarrow {B C}, \overrightarrow {C D}$ and $\overrightarrow {D A}$, which are the sides of this parallelogram.

Points $K$ and $L$ are the midpoints of the sides $B C$ and $C D$ of a parallelogram $A B C D$. Setting $\overrightarrow {A K} = \mathbf{k}$ and $\overrightarrow {A L} = \mathbf{l}$, express in terms of the vectors $\mathbf{k}$ and $\mathbf{l}$ the vectors $\overrightarrow {B C}$ and $\overrightarrow {C D}$.

The vectors $\overrightarrow {A B} = \mathbf{p}$ and $\overrightarrow {A F} = \mathbf{q}$ are two adjacent sides of a regular hexagon $A B C D E F$. Express in terms of $\mathbf{p}$ and $\mathbf{q}$ the vectors $\overrightarrow {B C}, \overrightarrow {C D}, \overrightarrow {D E}, \overrightarrow {E F}$ running along the sides of this hexagon.

In a triangle, find a point such that the sum of the vectors going from this point to the vertices of the triangle equals $\mathbf{0}$.

Two vectors $\overrightarrow {O A} = \mathbf{a}, \overrightarrow {O B} = \mathbf{b}$ issue from a point $O$. Find some vector $\overrightarrow {O M}$ lying along the bisector of the angle $A O B$. Deduce from this the angle bisector theorem.

A parallelepiped $A B C D A^{\prime } B^{\prime } C^{\prime } D^{\prime }$ is built on three noncoplanar vectors $\overrightarrow {A B} = \mathbf{p}, \overrightarrow {A D} = \mathbf{q}, \overrightarrow {A A^{\prime }} = \mathbf{r}$. Express in terms of $\mathbf{p}$, $\mathbf{q}$ and $\mathbf{r}$ the vectors coinciding with the edges, the diagonal of the parallelepiped, and the diagonals of the faces of this parallelepiped, for which the vertex $A^{\prime }$ serves as the initial point.

In a tetrahedron $A B C D$ the vectors of the edges issuing from the vertex $A$ are given: $\overrightarrow {A B} = \mathbf{b}, \overrightarrow {A C} = \mathbf{c}, \overrightarrow {A D} = \mathbf{d}$. Express in terms of them the vectors of the remaining edges of the tetrahedron, the median $D M$ of the face $B C D$, and the vector $\overrightarrow {A Q}$, where $Q$ is the point where the medians of the face $B C D$ meet.

A tetrahedron $O A B C$ is given. Setting $\overrightarrow {O A} = \mathbf{a}, \overrightarrow {O B} = \mathbf{b}, \overrightarrow {O C} = \mathbf{c}$, express in terms of $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ the vectors $\overrightarrow {M N}, \overrightarrow {P Q}$ and $\overrightarrow {R S}$, where $M, P$ and $R$ are the midpoints of the edges $O A, O B$ and $O C$, and $N, Q$ and $S$ are the midpoints of the corresponding opposite edges.

A skew quadrilateral $A B C D$ is given. The vectors $\overrightarrow {A B} = \mathbf{m}$ and $\overrightarrow {C D} = \mathbf{p}$ are known. Find the vector $\overrightarrow {E F}$ joining the midpoints of the diagonals $A C$ and $B D$.

Prove that the sum of the vectors going from the center of a regular polygon to its vertices equals zero.

Prove that the vector going from an arbitrary point of the plane to the center of a regular polygon is the arithmetic mean of the vectors going from this point to the vertices of the polygon.

Given the position vectors $\mathbf{r}_{A}, \mathbf{r}_{B}, \mathbf{r}_{D}$ and $\mathbf{r}_{A^{\prime }}$ of four vertices of a parallelepiped $A B C D A^{\prime } B^{\prime } C^{\prime } D^{\prime }$, find the position vectors of its four remaining vertices.

The position vectors $\overrightarrow {O A} = \mathbf{r}_{1}, \overrightarrow {O B} = \mathbf{r}_{2}$ and $\overrightarrow {O C} = \mathbf{r}_{3}$ are edges of a parallelepiped. Find the position vector of the point where the diagonal of the parallelepiped issuing from the vertex $O$ meets the plane passing through the vertices $A, B$ and $C$.

Three vectors $\mathbf{a} = (5,7,2), \mathbf{b} = (3,0,4), \mathbf{c} = (-6,1,-1)$ are given. Find the vectors:

Represent the vector $\mathbf{d}$ as a linear combination of the vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ in each of the following cases:

Determine in which of the following cases the triples of vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are linearly dependent, and, in the cases where this is possible, represent the vector $\mathbf{c}$ as a linear combination of the vectors $\mathbf{a}$ and $\mathbf{b}$:

Show that whatever the three vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ and the three numbers $\lambda , \mu , \nu$ may be, the vectors $\lambda \mathbf{a}-\mu \mathbf{b}, \nu \mathbf{b}-\lambda \mathbf{c}, \mu \mathbf{c}-\nu \mathbf{a}$ are coplanar.

Find the projection onto the plane $O y z$ of the vector $(1,3,5)$ parallel to the vector $(4,-1,3)$.

Find the projection of the vector $(7,5,-3)$ onto the plane determined by the vectors $(1,1,1)$ and $(2,-1,1)$, parallel to the vector $(2,-2,1)$.

Four vectors are given

$\mathbf{a} = (1,5,3), \quad \mathbf{b} = (6,-4,-2), \quad \mathbf{c} = (0,-5,7), \quad \mathbf{d} = (-20,27,-35)$.

Choose numbers $\alpha , \beta$ and $\gamma$ so that the vectors $\alpha \mathbf{a}, \beta \mathbf{b}, \gamma \mathbf{c}$ and $\mathbf{d}$ form a closed broken line, if the start of each subsequent vector is placed at the end of the preceding one.

Find the coordinates of the vertices of a parallelogram $A B C D$ in an affine coordinate system whose origin is at the center $O$ of this parallelogram, and whose basis vectors are $\overrightarrow {A B}, \overrightarrow {B O}$.

Find the affine coordinates of the vertices of a regular hexagon $A B C D E F$, taking the point $A$ as the origin and the pair of vectors $\overrightarrow {A B}, \overrightarrow {B C}$ as the basis.

In a trapezoid $A B C D$ the ratio of the lengths of the bases $A D$ and $B C$ equals $k$. Taking the vertex $A$ as the origin of coordinates, and the vectors $\overrightarrow {A D}$ and $\overrightarrow {A B}$ as the basis vectors, find the coordinates of the vertices of the trapezoid, of the point $M$ where the diagonals meet, and of the point $S$ where the extensions of the lateral sides meet.

Find the coordinates of the vertices of a tetrahedron $O A B C$ in a coordinate system with origin at the vertex $O$, whose basis vectors are the medians $\overrightarrow {O D}, \overrightarrow {O E}, \overrightarrow {O F}$ of the faces $B O C, C O A, A O B$.

Find the coordinates of the vertices of a tetrahedron $A B C D$, taking the point $P$ where the medians of the face $B C D$ meet as the origin, and the vectors $B Q, C R$, $D S$ as the basis vectors, where $Q, R$ and $S$ are the points where the medians of the faces $A C D, A B D$ and $A B C$ meet, respectively.

Find the coordinates of the center of gravity of a homogeneous rod $A O B$ bent at a right angle, if the lengths $\left|O A\right| = a$ and $\left|O B\right| = b$. Take the point $O$ as the origin of coordinates, and the vectors $\overrightarrow {O A}$ and $\overrightarrow {O B}$ as the unit vectors of the axes $O x$ and $O y$, respectively.

Find the center of gravity of a triangular wire frame.