Orientation; the Cross Product and Scalar Triple Product

A basis $\mathbf{e}_{1}, \mathbf{e}_{2}$ is given in the plane. Taking its orientation as positive, determine the orientation of the following bases:

Four nonzero vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ are given in the plane. It is known that the angles $(\widehat{\mathbf{a}, \mathbf{c}})$ and $(\widehat{\mathbf{b}, \mathbf{d}})$ are right, while $(\widehat{\mathbf{a}, \mathbf{d}})$ and $(\widehat{\mathbf{b}, \mathbf{c}})$ are acute. Do the pairs $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}, \mathbf{d}$ have the same or opposite orientation?

Using only the bilinearity and antisymmetry properties of signed area, prove that the ratio of the signed areas $S_{\text{or }}(\mathbf{u}, \mathbf{v})$ and $S_{\text{or }}(\mathbf{a}, \mathbf{b})$ equals

where

In which affine coordinate systems can formula (1.1) be used?

A vector $\mathbf{a} = (-5,2)$ is given. Find a vector $\mathbf{b}$ perpendicular to $\mathbf{a}$, equal to it in length, and directed so that the vectors $\mathbf{a}$ and $\mathbf{b}$ form a pair with the same orientation as the unit vectors of the axes $O x$ and $O y$.

Prove that

Using the properties of signed area, prove that the area of the triangle formed by the medians of a triangle $A B C$ equals $3/4$ of the area of the triangle $A B C$.

Three points in the plane are given: $A(-2,3), B(2,1), C(5,-3)$. Find the area $S$ of the triangle $A B C$ and the length $h$ of the altitude dropped onto the side $B C$.

The sides $B C, C A$ and $A B$ of a triangle $A B C$ are divided by points $P, Q, R$ in the ratios

Find the ratio of the area of the signed triangle $P Q R$ to the area of the signed triangle $A B C$.

The sides $B C, C A$ and $A B$ of a triangle $A B C$ are divided by points $P, Q, R$ in the ratios

Let $A^{\prime }, B^{\prime }, C^{\prime }$ be the points of intersection of the pairs of lines $B Q$ and $C R$, $C R$ and $A P$, $A P$ and $B Q$. Find the ratio of the area of the signed triangle $A^{\prime } B^{\prime } C^{\prime }$ to the area of the signed triangle $A B C$. What is this ratio equal to in the case $\lambda = \mu = \nu = 2$?

Find the angle from the vector $\mathbf{a}$ to the vector $\mathbf{b}$, if:

A basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ is given in space. Taking its orientation as positive, determine the orientation of the following bases:

Two bases $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ and $\mathbf{f}_{1}, \mathbf{f}_{2}, \mathbf{f}_{3}$ are given in space. It is known that the angle $(\widehat{\mathbf{e}_{i}, \mathbf{f}_{j}})$ is acute if $i = j$, and right otherwise.

Determine whether the given bases have the same or opposite orientation.

Four vectors $\overrightarrow {O A}(1,1,1), \overrightarrow {O B}(2,-1,2)$, $\overrightarrow {O C}(0,2,1)$ and $\overrightarrow {O D}(2,-2,1)$ are given in space. Does the ray $O D$ lie inside or outside the trihedral angle $O A B C$?

Find a vector $\overrightarrow {O M}$ forming equal acute angles with the rays $\overrightarrow {O A}(1,2,2), \overrightarrow {O B}(0,3,4)$ and $\overrightarrow {O C}(-2,-1,2)$. Does the ray $\overrightarrow {O M}$ lie inside or outside the trihedral angle $O A B C$?

Knowing the vector product $\mathbf{c}$ of the vectors $\mathbf{a}$ and $\mathbf{b}$, find:

Show that if three vectors $\mathbf{a}, \mathbf{b}$, $\mathbf{c}$ are not collinear, then the equalities $[\mathbf{a}, \mathbf{b}] = [\mathbf{b}, \mathbf{c}] = [\mathbf{c}, \mathbf{a}]$ are equivalent to the relation $\mathbf{a}+\mathbf{b}+\mathbf{c} = \mathbf{0}$.

Prove that if

then the vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are coplanar.

Three noncoplanar vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are drawn from one point. Show that the plane passing through the endpoints of these vectors is perpendicular to the vector $[\mathbf{a}, \mathbf{b}]+[\mathbf{b}, \mathbf{c}]+[\mathbf{c}, \mathbf{a}]$.

Show that if the vectors $[\mathbf{a}, \mathbf{b}],[\mathbf{b}, \mathbf{c}],[\mathbf{c}, \mathbf{a}]$ are coplanar, then they are collinear.

In an oriented space, two mutually perpendicular vectors $\mathbf{a}$ and $\mathbf{b}$ are given. Find the vector $\mathbf{c}$ obtained from the vector $\mathbf{b}$ by a rotation through $\pi / 2$ about the vector $\mathbf{a}$, such that the orientation of the triple of vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is positive.

In an oriented space, two nonzero vectors $\mathbf{a}$ and $\mathbf{b}$ are given. Find the vector $\mathbf{c}$ obtained from $\mathbf{b}$ by a rotation through an angle $\varphi$ in the positive direction about the vector $\mathbf{a}$, i.e., such that the triple $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is positive for $0 < \varphi < \pi$ and negative for $\pi < \varphi < 2 \pi$.

Find the vector product $[\mathbf{a}, \mathbf{b}]$ in each of the following cases:

Find the vector product $[\overrightarrow {A B}, \overrightarrow {A C}]$:

Compute the area of the parallelogram built on the vectors $\mathbf{a} = (8,4,1), \mathbf{b} = (2,-2,1)$.

Points in space are given: $A(2,-1,3), B(3,1,5), C(-1,3,2)$. Find the area $S$ of the triangle $A B C$ and the length $h$ of the altitude dropped onto the side $A B$.

The vectors $\mathbf{a} = (3,1,2), \mathbf{b} = (2,7,4), \mathbf{c} = (1,2,1)$ are given. Find:

Two vectors $\mathbf{a} = (11,10,2)$ and $\mathbf{b} = (4,0,3)$ are given. Find a vector $\mathbf{c}$ of length 1, perpendicular to the vectors $\mathbf{a}$ and $\mathbf{b}$ and directed so that the triple of vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ is positive.

Two vectors $\mathbf{a} = (1,1,1)$ and $\mathbf{b} = (1,0,0)$ are given. Find a unit vector $\mathbf{c}$, perpendicular to the vector $\mathbf{a}$, forming an angle $\pi / 3$ with the vector $\mathbf{b}$, and directed so that the triple of vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is positive.

Prove that the area of a convex plane quadrilateral $A B C D$ in space equals half the length of the vector product $[\overrightarrow {A C}, \overrightarrow {B D}]$.

In a regular tetrahedron $A B C D$, two cross-sections are drawn parallel to the edges $A C$ and $B D$. Find the edge length of the tetrahedron, if the areas of the cross-sections equal $S_{1}$ and $S_{2}$, and the distance between the cutting planes equals $d$.

Prove that for any two vectors $\mathbf{a}$ and $\mathbf{b}$ the equality $([\mathbf{a}, \mathbf{b}])^{2} = \mathbf{a}^{2} \mathbf{b}^{2}-(\mathbf{a}, \mathbf{b})^{2}$ holds.

Prove that for an arbitrary vector $\mathbf{a}$ and perpendicular vectors $\mathbf{b}$ and $\mathbf{c}$ the equality $[[\mathbf{a}, \mathbf{b}], \mathbf{c}] = \mathbf{b}(\mathbf{a}, \mathbf{c})$ holds.

Prove that for arbitrary vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ in space the equalities hold:

Prove that for arbitrary three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ the Jacobi identity holds

For any three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ prove the equality

Prove that for any four vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ the equalities

hold.

Prove the identities:

Prove that all four faces of an arbitrary tetrahedron have equal area if and only if they are congruent.

The vertices of a tetrahedron are given: $A(2,3,1), B(4,1,-2), C(6,3,7)$, $D(-5,-4,8)$. Find the length $h$ of its altitude dropped from the vertex $D$.

In a triangular pyramid, the edges issuing from the vertex $S$ have length 1, and the plane angles of the faces at the vertex $S$ equal $\pi / 4$, $\pi / 4$ and $\pi / 3$. Find the area of the base of the pyramid.

Prove that the equality $[[\mathbf{a}, \mathbf{b}], \mathbf{c}] = [\mathbf{a},[\mathbf{b}, \mathbf{c}]]$ holds if and only if at least one of the following two conditions is satisfied:

Three noncoplanar vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are given. Find the vector $\mathbf{x}$ satisfying the system of equations

The position vectors $\overrightarrow {O A} = \mathbf{r}_{1}, \overrightarrow {O B} = \mathbf{r}_{2}, \overrightarrow {O C} = \mathbf{r}_{3}$ of the points $A, B, C$ are given. It is known that the vectors $\mathbf{r}_{2}$ and $\mathbf{r}_{3}$ are not collinear. From the point $A$ a perpendicular $A M$ is dropped onto the plane $O B C$. Find the position vector of the point $M$.

Two triples of noncoplanar vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$ and $\mathbf{a}^{1}, \mathbf{a}^{2}, \mathbf{a}^{3}$ are called reciprocal if the vectors of these triples are related by

For a given triple of noncoplanar vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$, find the vectors $\mathbf{a}^{1}, \mathbf{a}^{2}, \mathbf{a}^{3}$ of its reciprocal triple.

The plane angles $\angle B O C = \alpha , \angle C O A = \beta , \angle A O B = \gamma$ of a trihedral angle $O A B C$ are given. Compute the cosines of its interior dihedral angles $A, B, C$, opposite to the faces $B O C, C O A, A O B$.

The interior dihedral angles $A, B, C$ are given. Compute the cosines of its plane angles $\alpha , \beta , \gamma$.

Prove that

(the law of sines in spherical geometry).

Compute the volume of a parallelepiped, knowing the lengths $a, b, c$ of three of its edges issuing from one vertex, and $\alpha , \beta , \gamma$ — the angles between them.

Prove the formulas of spherical geometry:

where $\alpha , \beta , \gamma$ are the side lengths of a triangle on the unit sphere; $A, B$, $C$ are the corresponding opposite angles.

Compute the volume of a triangular pyramid with lateral edges $a, b, c$ and dihedral angles $\alpha , \beta , \gamma$ at these edges.