Vectors and Coordinates in Space

The projections of a vector onto the axes $Ox$, $Oy$, $Oz$ are $(1,-4,8)$. Find the length of the vector and the cosines of its angles with the axes.

Find the sum of the vectors $\overrightarrow {a}+\overrightarrow {b}+\overrightarrow {c}$, if $\overrightarrow {a} = 5\overrightarrow {i}+7\overrightarrow {j}+8\overrightarrow {k}$, $\overrightarrow {b} = 3\overrightarrow {i}-4\overrightarrow {j}+6\overrightarrow {k}$, $\overrightarrow {c} = -6\overrightarrow {i}-9\overrightarrow {j}-5\overrightarrow {k}$, and the length of this vector.

Compute the scalar product $\overrightarrow {a} \cdot \overrightarrow {b}$, if $\overrightarrow {a} = 2\overrightarrow {i}+3\overrightarrow {j}+4\overrightarrow {k}$, $\overrightarrow {b} = 3\overrightarrow {i}+\overrightarrow {j}-3\overrightarrow {k}$.

Compute the vector product $\overrightarrow {a} \times \overrightarrow {b}$ of the vectors of the previous problem.

Verify the identities:

From the last identity of the previous problem, derive the Lagrange identity

Show that if $\overrightarrow {a}+\overrightarrow {b}+\overrightarrow {c} = 0$, then $\overrightarrow {a} \times \overrightarrow {b} = \overrightarrow {b} \times \overrightarrow {c} = \overrightarrow {c} \times \overrightarrow {a}$.

Find the angle between the vectors $\overrightarrow {i}+2\overrightarrow {j}+2\overrightarrow {k}$ and $-\overrightarrow {i}+2\overrightarrow {j}+4\overrightarrow {k}$.

Find the angle between the bisectors of the angles $xOy$ and $xOz$.

In triangle $ABC$ find the length of side $AB$ and the angle $BAC$, if the coordinates of the vertices are known: $A(2,-3,1)$, $B(4,11,6)$, $C(4,-4,3)$.

Using the formula for computing the vector product, find the area of triangle $ABC$, given the coordinates of the vertices $A(-2,1,3)$, $B(2,-1,7)$, $C(11,2,-5)$.

Find the projection of the vector $\overrightarrow {AB}$ onto the direction $CD$, if the points $A(1,-2,3)$, $B(4,-4,-3)$, $C(2,4,3)$, $D(8,6,6)$ are known.

Verify that the vectors $7\overrightarrow {i}+6\overrightarrow {j}-6\overrightarrow {k}$ and $6\overrightarrow {i}+2\overrightarrow {j}+9\overrightarrow {k}$ can be taken as edges of a cube. Find the third edge of the cube.

Compute the mixed (vector-scalar) product $(\overrightarrow {a} \times \overrightarrow {b}) \cdot \overrightarrow {c}$, if $\overrightarrow {a} = \overrightarrow {i}+3\overrightarrow {j}-\overrightarrow {k}$, $\overrightarrow {b} = 3\overrightarrow {i}+2\overrightarrow {j}$, $\overrightarrow {c} = 2\overrightarrow {i}-\overrightarrow {j}+3\overrightarrow {k}$.

Compute the following double vector products of the three vectors of the previous problem: $\overrightarrow {a} \times (\overrightarrow {b} \times \overrightarrow {c})$ and $(\overrightarrow {a} \times \overrightarrow {b}) \times \overrightarrow {c}$.

Compute the volume of tetrahedron $ABCD$, given the coordinates of its vertices $A(2,1,5)$, $B(4,0,8)$, $C(6,-2,6)$, $D(5,0,3)$.

Compute the altitude of tetrahedron $ABCD$ of the previous problem, dropped from the point $D$.

Determine the moment arm of the vector $\overrightarrow {AB}$ relative to the point $C$, i.e., the length of the perpendicular from the point $C$ to the line $AB$. The positions of the points $A(2,0,3)$, $B(4,6,0)$, $C(1,-3,8)$ are known.

Verify the identity

Prove the formula:

For what value of $m$ are the vectors $\overrightarrow {a} = \overrightarrow {i}+\overrightarrow {j}+m\overrightarrow {k}$; $\overrightarrow {b} = \overrightarrow {j}$; $\overrightarrow {c} = 3\overrightarrow {i}+\overrightarrow {k}$ coplanar, i.e. $(\overrightarrow {a} \times \overrightarrow {b}) \cdot \overrightarrow {c} = 0$?

Solve the previous problem, starting from the fact that when $\overrightarrow {a}, \overrightarrow {b}, \overrightarrow {c}$ are coplanar there is an equality $\overrightarrow {a} = \lambda \overrightarrow {b}+\mu \overrightarrow {c}$, where $\lambda$ and $\mu$ are scalar factors.

Three vectors $\overrightarrow {a}^{*}$, $\overrightarrow {b}^{*}$, $\overrightarrow {c}^{*}$ are called reciprocal to three noncoplanar vectors $\overrightarrow {a}$, $\overrightarrow {b}$, $\overrightarrow {c}$, if $\Delta \overrightarrow {a}^{*} = \overrightarrow {b} \times \overrightarrow {c}$, $\Delta \overrightarrow {b}^{*} = \overrightarrow {c} \times \overrightarrow {a}$, $\Delta \overrightarrow {c}^{*} = \overrightarrow {a} \times \overrightarrow {b}$, where the number $\Delta$ is defined by the equality $\Delta = (\overrightarrow {a} \times \overrightarrow {b}) \cdot \overrightarrow {c}$.

Prove that the vector $\overrightarrow {x}$, determined from the system of equations $\overrightarrow {x} \cdot \overrightarrow {a} = m$, $\overrightarrow {x} \cdot \overrightarrow {b} = n$, $\overrightarrow {x} \cdot \overrightarrow {c} = p$, where $\overrightarrow {a}$, $\overrightarrow {b}$, $\overrightarrow {c}$ and $m$, $n$, $p$ are given and $\Delta \neq 0$, has the form $\overrightarrow {x} = m\overrightarrow {a}^{*}+n\overrightarrow {b}^{*}+p\overrightarrow {c}^{*}$.

Represent in vector form the solution of the ordinary system

Verify that all vectors $\overrightarrow {x}$ satisfying the equation $\overrightarrow {x} \cdot \overrightarrow {a} = m$ can be given in the form: $\overrightarrow {x} = \dfrac {m}{\left|\overrightarrow {a}\right|^{2}}\overrightarrow {a}+\overrightarrow {a} \times \overrightarrow {c}$, where $\overrightarrow {c}$ is a completely arbitrary vector.

Verify that from the vector equation $\overrightarrow {x} \times \overrightarrow {a} = \overrightarrow {b}$ it is possible to find $\overrightarrow {x}$ only if $\overrightarrow {a}$ and $\overrightarrow {b}$ are mutually perpendicular, and in this case

where $n$ is an arbitrary scalar factor.

If the vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ are given, with $\overrightarrow {a}$ perpendicular to $\overrightarrow {b}$, then from the two vector equations $\overrightarrow {x} \cdot \overrightarrow {a} = m$ and $\overrightarrow {x} \times \overrightarrow {a} = \overrightarrow {b}$ the vector $\overrightarrow {x}$ is determined uniquely. Verify this and find $\overrightarrow {x}$.

Determine the vector $\overrightarrow {x}$ from the system $\overrightarrow {x} \cdot \overrightarrow {i} = 3$, $\overrightarrow {x} \times \overrightarrow {i} = -2\overrightarrow {k}$.

If a trihedral angle is cut by a sphere of unit radius centered at the vertex of the trihedral angle, then a spherical triangle is obtained on the surface of the sphere at the intersection. Its sides $\alpha$, $\beta$, $\gamma$ measure the plane angles of the trihedral angle, and its angles $A$, $B$, $C$ measure the dihedral angles of the trihedral angle.

Placing the unit vectors of the edges: a) $\overrightarrow {e}_{1}$ along the axis $Oz$, b) $\overrightarrow {e}_{2}$ in the plane $xOz$ at an angle $\alpha$ to the axis $Oz$, c) $\overrightarrow {e}_{3}$ in some meridional plane, and computing $\overrightarrow {e}_{2} \cdot \overrightarrow {e}_{3}$ in two ways, obtain the law of cosines of spherical trigonometry

With the same arrangement of unit vectors, compute $(\overrightarrow {e}_{1} \times \overrightarrow {e}_{2}) \cdot \overrightarrow {e}_{3}$. By cyclically permuting the unit vectors, obtain the law of sines of spherical trigonometry $\dfrac {\sin \alpha }{\sin A} = \dfrac {\sin \beta }{\sin B} = \dfrac {\sin \gamma }{\sin C}$.

Three unit vectors make angles $\alpha$, $\beta$, $\gamma$ with one another. Verify that the volume of the parallelepiped built on them (the sine of Staudt) equals

Find the volume of a parallelepiped whose edges have lengths $2, 3, 4$, with the plane angles of the trihedral angle $120^{\circ }$, $150^{\circ }$, $60^{\circ }$.

Verify that if $\overrightarrow {a}$ and $\overrightarrow {b}$ are any two nonparallel vectors, then the vectors $\dfrac {\overrightarrow {a}}{\left|\overrightarrow {a}\right|}$, $\dfrac {\overrightarrow {a} \times \overrightarrow {b}}{\left|\overrightarrow {a} \times \overrightarrow {b}\right|}$, $\dfrac {(\overrightarrow {a} \times \overrightarrow {b}) \times \overrightarrow {a}}{\left|(\overrightarrow {a} \times \overrightarrow {b}) \times \overrightarrow {a}\right|}$ are three mutually perpendicular unit vectors. Find their position relative to the given vectors.

The unit vectors $\overrightarrow {e}_{1}$ and $\overrightarrow {e}_{2}$ are given. Find the unit vector lying in the plane of $\overrightarrow {e}_{1}$ and $\overrightarrow {e}_{2}$ and making an angle $\theta$ with the vector $\overrightarrow {e}_{1}$.

Find the unit vector lying in the plane of the vectors $\overrightarrow {r}_{1} = 6\overrightarrow {i}+2\overrightarrow {j}+3\overrightarrow {k}$ and $\overrightarrow {r}_{2} = 6\overrightarrow {i}-2\overrightarrow {j}+3\overrightarrow {k}$, between them, and making an angle of $45^{\circ }$ with $\overrightarrow {r}_{1}$.

Rotate the vector $\overrightarrow {OM} = \overrightarrow {r}$ through an angle $\omega$ about an axis having the direction of the unit vector $\overrightarrow {e}$ and passing through the point $O$.

Rotate the vector $\overrightarrow {OM} = 2\overrightarrow {i}+\overrightarrow {j}+2\overrightarrow {k}$ about the bisector of the angle $xOy$ through $60^{\circ }$.

Prove the equality:

where the letters $\alpha$, $\beta$, $\gamma$ denote the cosines of the angles between the old and new axes.

The cosines of the angles between the new and old axes are given by the table:

Prove that for points for which $x : y : z = 4 : (-2) : 1$, the new coordinates equal the old ones.