Linear relations

Prove the following statements:

Can a system consisting of a single vector be linearly dependent?

Prove that for any three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and any three numbers $\alpha , \beta , \gamma$ the vectors $\alpha \mathbf{a} - \beta \mathbf{b}, \gamma \mathbf{b} - \alpha \mathbf{c}, \beta \mathbf{c} - \gamma \mathbf{a}$ are linearly dependent.

Given three vectors $\mathbf{a}(1, 2), \mathbf{b}(- 5, - 1)$, $\mathbf{c}(- 1, 3)$. Find the coordinates of the vectors $2\mathbf{a} + 3\mathbf{b} - \mathbf{c}, 16\mathbf{a} + 5\mathbf{b} - 9\mathbf{c}$.

Given three vectors $\mathbf{a}(1, 3), \mathbf{b}(2, - 1), \mathbf{c}(- 4, 1)$. Find numbers $\alpha$ and $\beta$ such that $\alpha \mathbf{a} + \beta \mathbf{b} + \mathbf{c} = \mathbf{o}$.

Verify that the vectors $\mathbf{a}(- 5, - 1)$ and $\mathbf{b}(- 1, 3)$ form a basis in the plane. Find the coordinates of the vectors $\mathbf{c}(- 1, 2)$ and $\mathbf{d}(2, - 6)$ in this basis.

In a certain basis, vector $\mathbf{a}$ has coordinates $(x$, $1 - x)$, and vector $\mathbf{b}$ has coordinates $\left(x^{2} - 2x, x^{2} - 2x + 1\right)$. For which values of $x$ are the vectors

Given four vectors $\mathbf{a}(3, 0, - 2), \mathbf{b}(1, 2, - 5), \mathbf{c}(- 1, 1, 1)$, $\mathbf{d}(8, 4, 1)$. Find the coordinates of the vectors $- 5\mathbf{a} + \mathbf{b} - 6\mathbf{c} + \mathbf{d}, 3\mathbf{a} - \mathbf{b} - \mathbf{c} - \mathbf{d}$.

Given four vectors $\mathbf{a}(4, 1, - 1), \mathbf{b}(3, - 1, 0), \mathbf{c}(- 1, 1, 1)$, $\mathbf{d}(- 1, 3, 4)$. Find numbers $\alpha , \beta , \gamma$ such that $\alpha \mathbf{a} + \beta \mathbf{b} + \gamma \mathbf{c} + \mathbf{d} = \mathbf{o}$.

Verify that the vectors $\mathbf{a}(4, 1, - 1), \mathbf{b}(1, 2, - 5)$ and $\mathbf{c}(- 1, 1, 1)$ form a basis in space. Find the coordinates of the vectors $\mathbf{l}(4, 4, - 5), \mathbf{m}(2, 4, - 10), \mathbf{n}(0, 3, - 4)$ in this basis.

Check whether the vectors $\mathbf{l}, \mathbf{m}$ and n are coplanar; if the answer is affirmative, state the linear relation connecting them (here $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three noncoplanar vectors):

From one point in space, three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are drawn. Prove that the endpoint of vector $\mathbf{c}$ lies on the segment joining the endpoints of vectors $\mathbf{a}$ and $\mathbf{b}$ if and only if the equality $\mathbf{c} = \alpha \mathbf{a} + \beta \mathbf{b}$ holds, where $\alpha \geqslant 0, \beta \geqslant 0, \alpha + \beta = 1$. In what ratio does the endpoint of vector $\mathbf{c}$ divide this segment?

In parallelogram $ABCD$, point $K$ is the midpoint of segment $BC$ and point $O$ is the intersection point of the diagonals. Taking $\overrightarrow {AB}$ and $\overrightarrow {AD}$ as basis vectors, find in this basis the coordinates of the vectors $\overrightarrow {BD}, \overrightarrow {CO}, \overrightarrow {KD}$.

In triangle $ABC$, point $M$ is the midpoint of segment $AB$ and point $O$ is the intersection point of the medians. Taking $\overrightarrow {AB}$ and $\overrightarrow {AC}$ as basis vectors, find in this basis the coordinates of the vectors $\overrightarrow {AM}, \overrightarrow {AO}, \overrightarrow {MO}$.

In trapezoid $ABCD$ the lengths of the bases $AD$ and $BC$ are in the ratio $3: 2$. Taking $\overrightarrow {AC}$ and $\overrightarrow {BD}$ as basis vectors, find in this basis the coordinates of the vectors $\overrightarrow {AB}, \overrightarrow {BC}, \overrightarrow {CD}, \overrightarrow {DA}$.

In trapezoid $ABCD$ the lengths of the bases $AD$ and $BC$ are in the ratio $3: 1. O$ is the intersection point of the diagonals of the trapezoid, $S$ is the intersection point of the extensions of the lateral sides. Taking $\overrightarrow {AD}$ and $\overrightarrow {AB}$ as basis vectors, find the coordinates of the vectors $\overrightarrow {AC}, \overrightarrow {AO}, \overrightarrow {AS}$.

Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$ of quadrilateral $ABCD$. Prove that $\overrightarrow {EF} = (\overrightarrow {BC} + \overrightarrow {AD}) / 2$.

Given a regular hexagon $ABCDEF$. Taking $\overrightarrow {AB}$ and $\overrightarrow {AE}$ as basis vectors, find in this basis the coordinates of the vectors $\overrightarrow {BC}, \overrightarrow {CD}, \overrightarrow {DE}, \overrightarrow {EF}, \overrightarrow {BD}, \overrightarrow {CF}, \overrightarrow {CE}$.

In the trapezoid of Problem 1.16, point $M$ is the midpoint of side $CD$. Find the coordinates of the vector $\overrightarrow {AD}$ in the basis $\overrightarrow {OS}, \overrightarrow {OM}$.

In triangle $ABC$, points $K$ and $L$ are the midpoints of sides $BC$ and $AC$ respectively. Points $M$ and $N$ lie respectively on segments $AK$ and $BL$ so that $\left|AM\right|:\left|MK\right| = 6: 1$ and $\left|BN\right|:\left|NL\right| = 8: 1$. Point $P$ is the midpoint of segment $MN$. Find the coordinates of the vector $\overrightarrow {AB}$ in the basis $\overrightarrow {MN}, \overrightarrow {CP}$.

In triangle $ABC$, point $M$ is the midpoint of side $AC$, points $K$ and $L$ on sides $AB$ and $BC$ are located so that $\left|AK\right|:\left|KB\right| = 3: 5$, and $\left|BL\right|:\left|LC\right| = 2: 3$. Find the coordinates of the vector $\overrightarrow {BM}$ in the basis $\overrightarrow {AL}, \overrightarrow {CK}$.

In triangle $ABC$, points $K, L, M$ are located respectively on sides $AB, BC$ and $AC$ so that $\left|AK\right|$: $:\left|KB\right| = \left|BL\right|:\left|LC\right| = \left|CM\right|:\left|MA\right| = 3: 1$. The medians of triangle $ABC$ meet at point $P$. Find the coordinates of the vector $\overrightarrow {AP}$ in the basis $\overline{LK}, \overrightarrow {LM}$.

In tetrahedron $OABC$, points $K, L, M, N, P, Q$ are the midpoints of edges $OA, OB, OC, AB, AC, BC$ respectively, and $S$ is the intersection point of the medians of triangle $ABC$. Taking $\overrightarrow {OA}, \overrightarrow {OB}$ and $\overrightarrow {OC}$ as basis vectors, find in this basis the coordinates of:

Given three points $O, A, B$ not lying on a single line. Taking $\overrightarrow {OA}$ and $\overrightarrow {OB}$ as basis vectors, find:

In triangle $ABC$, the bisector $AD$ is drawn. Find the coordinates of the vector $\overrightarrow {AD}$ in the basis formed by the vectors $\overrightarrow {AB}$ and $\overrightarrow {AC}$.

Given a regular hexagon $ABCDEF$. Taking vertex $A$ as the origin, and $\overrightarrow {AC}$ and $\overrightarrow {AE}$ as basis vectors, find the coordinates of the vertices of the hexagon and its center.

In trapezoid $ABCD$ the ratio of the lengths of the bases $AD$ and $BC$ is 4. Taking vertex $A$ as the origin, and $\overrightarrow {AD}$ and $\overrightarrow {AB}$ as basis vectors, find the coordinates of the vertices of the trapezoid, of the point $M$ where its diagonals meet, and of the point $S$ where the lateral sides meet.

Given a parallelepiped $ABCDA_{1} B_{1} C_{1} D_{1}$. Taking vertex $A$ as the origin, and $\overrightarrow {AB}, \overrightarrow {AD}$ and $\overrightarrow {AA_{1}}$ as basis vectors, find the coordinates of:

Three points $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)$, not lying on a single line, are consecutive vertices of a parallelogram. Find the coordinates of the fourth vertex $D$ of this parallelogram.

Given two distinct points $A\left(x_{1}, y_{1}, z_{1}\right), B\left(x_{2}, y_{2}, z_{2}\right)$. Find the coordinates of:

Given two points $A(3, - 2)$ and $B(1, 4)$. Point $M$ lies on line $AB$, and $\left|AM\right| = 3\left|AB\right|$. Find the coordinates of point $M$, if:

Given three points $A\left(x_{1}, y_{1}, z_{1}\right), B\left(x_{2}, y_{2}, z_{2}\right), C\left(x_{3}, y_{3}, z_{3}\right)$, not lying on a single line. Find the coordinates of the point where the medians of triangle $ABC$ meet.

Given the position vectors $\mathbf{r}_{1}, \mathbf{r}_{2}, \mathbf{r}_{3}, \mathbf{r}_{4}$ of the vertices $A, B$, $D, A_{1}$ of a parallelepiped $ABCDA_{1} B_{1} C_{1} D_{1}$, express through them the position vectors of the remaining four vertices.

The ratio of the lengths of the bases $AD$ and $BC$ of trapezoid $ABCD$ is $m: n$. Express the position vectors of vertex $D$, of the point $M$ where the diagonals of the trapezoid meet, and of the point $S$ where the lateral sides meet, through the position vectors $\mathbf{r}_{1}, \mathbf{r}_{2}, \mathbf{r}_{3}$ of vertices $A, B, C$.

Prove that the position vector of the center of a regular polygon is the arithmetic mean of the position vectors of its vertices.

Given the position vectors $\mathbf{r}_{1}, \mathbf{r}_{2}, \mathbf{r}_{3}$ of the vertices of a triangle, find the position vector of the center of the circle inscribed in the triangle.

In the plane of triangle $ABC$, find a point $O$ such that $\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} =$ o. Do such points exist outside the plane of the triangle?

Masses $m_{1}, \ldots , m_{n}$ are concentrated at points with position vectors $\mathbf{r}_{1}, \ldots , \mathbf{r}_{n}$. Find the position vector of the center of mass of this system.

A homogeneous wire is bent into the shape of an angle $AOB$ with sides $\left|OA\right| = a$ and $\left|OB\right| = b$. Find the coordinates of the center of mass of the wire in the coordinate system $O, \overrightarrow {OA} / a, \overrightarrow {OB} / b$.

Find the coordinates of the center of mass of a homogeneous plate in the shape of quadrilateral $ABCD$ with vertices at the points $A(3, 1), B(7, 3), C(0, 4), D(- 1, 2)$.

Prove that if the diagonals of a quadrilateral bisect each other at their point of intersection, then this quadrilateral is a parallelogram.

Points $K$ and $L$ are the midpoints of sides $AB$ and $BC$ of parallelogram $OABC$. Prove that the intersection point of the diagonals of $OABC$ coincides with the intersection point of the medians of triangle $OKL$.

Point $K$ lies on the extension of side $AB$ of triangle $ABC$ beyond point $B$, point $L$ lies on the extension of side $BC$ beyond point $C$, point $M$ lies on the extension of side $CA$ beyond point $A$, and $\left|AB\right|:\left|BK\right| = \left|BC\right|:\left|CL\right| = \left|CA\right|:\left|AM\right|$. Prove that the intersection points of the medians of triangles $ABC$ and $KLM$ coincide.

On sides $AB$ and $AC$ of triangle $ABC$, points $M$ and $N$ are taken respectively so that $\left|AM\right|:\left|BM\right| = m_{1}: n_{1}$, $\left|AN\right|:\left|CN\right| = m_{2}: n_{2}$. Denote by $O$ the intersection point of segments $BN$ and $CM$. Find the ratios $\left|BO\right|:\left|ON\right|$ and $\left|CO\right|:\left|OM\right|$.

Applying the result of Problem 1.44 with $m_{1} = n_{1} = m_{2} = n_{2} = 1$, prove that the medians of a triangle meet at a single point.

Vertex $D$ of parallelogram $ABCD$ is joined to point $K$ lying on side $BC$ such that $\left|BK\right|:\left|KC\right| = 2: 3$. Vertex $B$ is joined to point $L$ lying on side $CD$ such that $\left|CL\right|:\left|LD\right| = 5: 3$. In what ratio does the point $M$ where lines $DK$ and $BL$ meet divide segments $DK$ and $BL$?

On the legs $AB$ and $BC$ of isosceles triangle $ABC$, points $M$ and $N$ are located respectively so that $\left|AM\right|:\left|BM\right| = m: 1, \left|CN\right|:\left|BN\right| = n: 1$. Line $MN$ meets the altitude $BD$ of the triangle at point $O$. Find the ratio $\left|DO\right|:\left|BO\right|$.

On sides $AB, BC$ and $CA$ of triangle $ABC$, points $M, N, P$ are taken respectively so that $\left|AM\right| = \left|AB\right| / n, \left|BN\right| = \left|BC\right| / n, \left|CP\right| = \left|CA\right| / n$. The area of triangle $ABC$ equals $S$. Find the area of the triangle formed by the intersection of lines $AN, BP$ and $CM$. Deduce from this that the medians of a triangle meet at a single point.

Prove that the four segments joining the vertices of a tetrahedron to the points where the medians of the opposite faces meet, intersect at a single point and are divided at this point in the ratio $3: 1$, measured from the vertex.

Prove that the three segments joining the midpoints of skew edges of a tetrahedron intersect at a single point and are bisected at this point.

On the diagonals $AB_{1}$ and $CA_{1}$ of the lateral faces of triangular prism $ABCA_{1} B_{1} C_{1}$, points $E$ and $F$ are located respectively so that lines $EF$ and $BC_{1}$ are parallel. Find the ratio $\left|EF\right|:\left|BC_{1}\right|$.

On the diagonal $BC_{1}$ of a lateral face of triangular prism $ABCA_{1} B_{1} C_{1}$ a point $M$ is taken, and on the diagonal $CA_{1}$ of another lateral face, a point $N$. Line $MN$ is parallel to plane $ABB_{1} A_{1}$. Find the ratio $\left|CN\right|:\left|CA_{1}\right|$, if $\left|BM\right|:\left|BC_{1}\right| = 1: 3$.