The scalar product of vectors

Find the dot product of the vectors $\mathbf{a}$ and $\mathbf{b}$, if:

Compute the expression $\left|\mathbf{a}\right|^{2} - \sqrt{3}(\mathbf{a}, \mathbf{b}) + 5\left|\mathbf{b}\right|^{2}$, if:

Find the dot product of the vectors $\mathbf{a}$ and $\mathbf{b}$ given by their coordinates:

Find the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ given by their coordinates:

Find the distance between the points $A$ and $B$ given by their coordinates:

Find the dot product of the vectors $\mathbf{a}$ and $\mathbf{b}$ given by their coordinates:

Find the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ given by their coordinates:

Find the distance between the points $A$ and $B$ given by their coordinates:

Given three vectors: $\mathbf{a}(- 1, 2), \mathbf{b}(5, 1), \mathbf{c}(4, - 2)$. Compute:

Given three vectors: $\mathbf{a}(1, - 1, 1), \mathbf{b}(5, 1, 1), \mathbf{c}(0, 3, - 2)$. Compute:

Prove that the vectors $\mathbf{a}$ and $\mathbf{b}(\mathbf{a}, \mathbf{c}) - \mathbf{c}(\mathbf{a}, \mathbf{b})$ are mutually perpendicular.

Is it true that for any vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ the relation $(\mathbf{a}, \mathbf{b}) \cdot (\mathbf{c}, \mathbf{d}) = (\mathbf{a}, \mathbf{c}) \cdot (\mathbf{b}, \mathbf{d})$ holds?

Given three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ such that $\left|\mathbf{a}\right| = \left|\mathbf{b}\right| = \left|\mathbf{c}\right| = 1$, $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{o}$. Compute $(\mathbf{a}, \mathbf{b}) + (\mathbf{b}, \mathbf{c}) + (\mathbf{c}, \mathbf{a})$.

In triangle $ABC$ the lengths of the sides are given. Find the dot product $(\overrightarrow {AC}, \overrightarrow {BC})$, if:

Given a triangle $ABC$. Express through $\mathbf{b} = \overrightarrow {AB}$ and $\mathbf{c} = \overrightarrow {AC}$

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

Prove that for an arbitrary rectangle $ABCD$ and an arbitrary point $M$ (lying or not lying in the plane of the rectangle) the following equalities hold:

In trapezoid $ABCD$ the ratio of the lengths of the bases $\left|AD\right|$: $:\left|BC\right|$ is 3. Express through $\mathbf{b} = \overrightarrow {AB}$ and $\mathbf{c} = \overrightarrow {AC}$:

The lengths of the basis vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ of a general Cartesian coordinate system in the plane are $\sqrt{2}$ and 1 respectively, and the angle between them is $45^{\circ }$. Compute the lengths of the diagonals and the angles of the parallelogram built on the vectors having coordinates $(2, 2)$ and $(- 1, 4)$ in this basis.

The lengths of the basis vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ of a general Cartesian coordinate system in the plane are 4 and 2 respectively, and the angle between the basis vectors is $120^{\circ }$. Relative to this coordinate system the vertices of triangle $A(- 2, 2), B(- 2, - 1), C(- 1, 0)$ are given. Find the lengths of the sides and the angles of the triangle.

The lengths of the basis vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ are $3, \sqrt{2}, 4$ respectively, and the angles between them are $\angle \left(\mathbf{e}_{1}, \mathbf{e}_{2}\right) = \angle \left(\mathbf{e}_{2}, \mathbf{e}_{3}\right) = 45^{\circ }, \angle \left(\mathbf{e}_{1}, \mathbf{e}_{3}\right) = 60^{\circ }$. Compute the lengths of the sides and the angles of the parallelogram built on the vectors having coordinates $(1, - 3, 0)$ and $(- 1, 2, 1)$ in this basis.

The lengths of the basis vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ are $1, 1, 2$ respectively; the angles between them are $\angle \left(\mathbf{e}_{1}, \mathbf{e}_{2}\right) = 90^{\circ }$, $\angle \left(\mathbf{e}_{1}, \mathbf{e}_{3}\right) = \angle \left(\mathbf{e}_{2}, \mathbf{e}_{3}\right) = 60^{\circ }$. Compute the area of the parallelogram built on the vectors $\mathbf{a}(- 1, 0, 2)$ and $\mathbf{b}(2, - 1, 1)$.

From one point, three vectors $\mathbf{a}(0, - 3, 4)$, $\mathbf{b}(4, 1, - 8)$ and $\mathbf{c}$ are drawn. Vector $\mathbf{c}$ has length 1 and bisects the angle between $\mathbf{a}$ and b. Compute the coordinates of the vector $\mathbf{c}$.

Given two vectors $\mathbf{a}$ and $\mathbf{b}$, with $\mathbf{a} \neq \mathbf{o}$. Express through $\mathbf{a}$ and $\mathbf{b}$ the orthogonal projection of the vector $\mathbf{b}$ onto the line whose direction is determined by the vector $\mathbf{a}$.

Find the sum of the orthogonal projections of the vector $\mathbf{a}$ onto the sides of a regular triangle.

Given a vector $\mathbf{a}(1, 1)$. Find the orthogonal projection of the vector $\mathbf{b}$ onto the line whose direction is determined by the vector $\mathbf{a}$, and the orthogonal component of the vector $\mathbf{b}$ relative to this line, if the vector $\mathbf{b}$ has coordinates:

Given a vector $\mathbf{a}$ $(1, - 1, 2)$. Find the orthogonal projection of the vector $\mathbf{b}$ onto the line whose direction is determined by the vector $\mathbf{a}$, and the orthogonal component of the vector $\mathbf{b}$ relative to this line, if the vector $\mathbf{b}$ has coordinates:

Given two vectors $\mathbf{a}(3, - 1)$ and $\mathbf{b}(- 1, 1)$. Find the vector $\mathbf{x}$ satisfying the system of equations $(\mathbf{x}, \mathbf{a}) = 13$, $(\mathbf{x}, \mathbf{b}) = - 3$.

Given vectors $\mathbf{a}(\sqrt{3}, - 3)$ and $\mathbf{b}(1, - 1)$. Find all vectors $\mathbf{x}$ that make an angle $\pi / 3$ with the vector $\mathbf{a}$ and are such that $(\mathbf{b}, \mathbf{x}) = 1$.

Given three vectors $\mathbf{a}(4, 1, 5), \mathbf{b}(0, 5, 2)$ and $\mathbf{c}(- 6, 2, 3)$. Find the vector $\mathbf{x}$ satisfying the system of equations $(\mathbf{x}, \mathbf{a}) = 18, (\mathbf{x}, \mathbf{b}) = 1, (\mathbf{x}, \mathbf{c}) = 1$.

Given a nonzero vector $\mathbf{a}$ and a scalar $p$. Express through $\mathbf{a}$ and $p$ some vector $\mathbf{x}$ satisfying the equation $(\mathbf{x}, \mathbf{a}) = p$.

Explain the geometric meaning of all solutions of the vector equation $(\mathbf{x}, \mathbf{a}) = p$, as well as of its particular solution collinear to the vector $\mathbf{a}$:

Explain the geometric meaning of:

Given two vectors $\mathbf{a}(1, - 1, 1)$ and $\mathbf{b}(5, 1, 1)$. Compute the coordinates of a vector $\mathbf{c}$ that has length 1 and is orthogonal to the vectors $\mathbf{a}$ and $\mathbf{b}$. How many solutions does the problem have?

Given two vectors $\mathbf{a}(1, - 1, 1)$ and $\mathbf{b}(5, 1, 1)$. Vector $\mathbf{c}$ has length 1, is orthogonal to the vector $\mathbf{a}$, and makes an angle $\arccos (\sqrt{2 / 27})$ with the vector $\mathbf{b}$. Compute the coordinates of the vector $\mathbf{c}$. How many solutions does the problem have?

In an isosceles triangle, the medians drawn to the legs are mutually perpendicular. Find the angles of the triangle.

In parallelogram $ABCD$, points $K$ and $L$ are the midpoints of sides $BC$ and $CD$. Find $\left|AD\right|$, if $\left|AK\right| = 6, \left|AL\right| = 3$, and angle $KAL = \pi / 3$.

The lengths of the sides of a triangle are related by $a^{2} + b^{2} = 5c^{2}$. Prove that two medians of the triangle are perpendicular.

The lengths of adjacent sides of a parallelogram are in the ratio $m: n$, and the angle between these sides equals $\alpha$. Find the angle between the diagonals of the parallelogram.

In a convex quadrilateral, the sum of the squares of two opposite sides equals the sum of the squares of the other two opposite sides. Find the angle between the diagonals of the quadrilateral.

In a right trapezoid, the diagonals are mutually perpendicular, and the ratio of the lengths of the bases equals $m: n(m > n$). Find:

Prove that if the lengths of two medians, the lengths of two altitudes, or the lengths of two bisectors of a triangle are equal, then this triangle is isosceles.

Let $M$ be the point where the medians of triangle $ABC$ meet. Prove that $\left|AM\right|^{2} + \left|BM\right|^{2} + \left|CM\right|^{2} = \left(\left|AB\right|^{2} + + \left|BC\right|^{2} + \left|AC\right|^{2}\right) / 3$.

The lengths of edges $AA_{1}, AB$ and $AD$ of parallelepiped $ABCDA_{1} B_{1} C_{1} D_{1}$ are $a, b, c$ respectively. The angles between them $\angle BAD, \angle A_{1} AD$ and $\angle A_{1} AB$ are $\alpha , \beta , \gamma$ respectively. Find the length of the diagonal $AC_{1}$.

Given an arbitrary tetrahedron $ABCD$. Prove: if edges $AB$ and $CD$ are perpendicular and edges $AC$ and $BD$ are perpendicular, then edges $BC$ and $AD$ are also perpendicular.

Given two segments $AB$ and $CD$ (in space, generally speaking). Prove that the segments are perpendicular if $\left|AC\right|^{2} + \left|BD\right|^{2} = \left|AD\right|^{2} + \left|BC\right|^{2}$. Is the converse statement true?

In a regular tetrahedron $ABCD$, points $M$ and $P$ are the midpoints of edges $AD$ and $CD$ respectively, and points $N$ and $Q$ are the centers of faces $BCD$ and $ABC$ respectively. Find the angle between lines $MN$ and $PQ$.

The edge length of cube $ABCDA_{1} B_{1} C_{1} D_{1}$ equals $a$. Point $P$ is the midpoint of edge $CC_{1}$, point $Q$ is the center of face $AA_{1} B_{1} B$. Segment $MN$ with endpoints on lines $AD$ and $A_{1} B_{1}$ meets line $PQ$ and is perpendicular to it. Find the length of this segment.

In a regular tetrahedron $ABCD$, points $E$ and $F$ are the midpoints of edges $AD$ and $BC$ respectively. On edge $CD$ a point $N$ is taken, and on segment $EF$ a point $M$, so that $\angle MNC = 45^{\circ }, \angle NME = \arccos (2 / 3)$. In what ratio do points $M$ and $N$ divide segments $EF$ and $CD$?

In a regular hexagonal pyramid $SABCDEF$ (S is the apex), the side length of the base equals 2. Vertices $K$ and $M$ of rhombus $KLMF$ lie on edges $AB$ and $SD$ respectively, and $\left|KM\right| = 3$, while segment $KL$ meets edge $SB$. Find the volume of the pyramid.