Example 4. Find the coordinates of the vector $\vec{x}$, if $\vec{x} \perp \vec{c}_{1}, \vec{x} \perp \vec{c}_{2}$, where $\overrightarrow{c_{1}}=\{2,-3,2\}, \overrightarrow{c_{2}}=\{-1,3,0\}$ and $|\vec{x}|=7$.
Solution. In Fig. 3.9, we have $\vec{x} \perp \vec{c}_{1}, \vec{x} \perp \vec{c}_{2}$.

From the conditions of vector perpendicularity (point $2^{\circ}$), we have $\vec{x} \cdot \vec{c}_{1}=0 ; \vec{x} \cdot \vec{c}_{2}=0$. Let $\vec{x}=\{x, y, z\}$. The condition of the problem can be rewritten as a system:

$$
\left\{\begin{array}{r}
2 x - 3 y + 2 z = 0, \\
- x + 3 y = 0, \\
x^{2} + y^{2} + z^{2} = 49
\end{array} \Rightarrow \left\{\begin{array}{l}
x = 3 y \\
z = - \frac{3}{2} y, \\
9 y^{2} + y^{2} + \frac{9}{4} y^{2} = 49
\end{array} \Rightarrow \left\{\begin{array}{l}
y = \pm 2 \\
x = \pm 6 \\
z = \mp 3
\end{array}\right.\right.\right.
$$

Answer. $\vec{x}=\{ \pm 6, \pm 2, \mp 3\}$.

## Exercises

1. Find the unit vector perpendicular to the vectors $\vec{a}=\{1,0,2\}$ and $\vec{b}=\{0,-1,2\}$.
2. Vectors $\vec{a}, \vec{b}, \vec{c}$ have equal lengths and form equal angles pairwise. Find the vector $\vec{c}$, if $\vec{a}=\vec{i}+\vec{j}, \vec{b}=\vec{j}+\vec{k}$.
3. For what value of $m$ are the vectors $\vec{a}=\{m, 1,0\}$ and $\vec{b}=\{3,-3,4\}$ perpendicular?
4. Find the scalar product of the vectors $3 \vec{a}-2 \vec{b}$ and $5 \vec{a}-6 \vec{b}$, if $|\vec{a}|=4,|\vec{b}|=6$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$.
5. In triangle $A B C$, whose vertices lie at points $A(1,1,-1), B(2,3,1), C(3,2,1)$, find: 1) the internal angles; 2) the lengths of the sides; 3) the acute angle between the median $B D$ and the side $A C$.
6. Find the projection of the vector $\overrightarrow{C D}=\{-1,4,-9\}$ onto the direction of the vector $\overrightarrow{A D}=\{4,12,-6\}$.
7. Find the angle between the diagonals of the parallelogram constructed on the vectors $\vec{a}=2 \vec{i}+\vec{j}$ and $\vec{b}=-2 \vec{j}+\vec{k}$.
8. Calculate $3 \vec{a}^{2}-2 \vec{a} \vec{b}+4 \vec{b}^{2}$, if $a=\frac{1}{3}, b=6,(\widehat{a}, \vec{b})=60^{\circ}$.
9. Prove that the diagonals of a rhombus constructed on the vectors $\vec{a}$ and $\vec{b}$ are mutually perpendicular.
10. Calculate $\vec{a}^{2}+3 \vec{a} \vec{b}-2 \vec{b} \vec{c}+10$, if $\vec{a}=4 \vec{m}-\vec{n}, \vec{b}=\vec{m}+2 \vec{n}$, $\vec{c}=2 \vec{m}-3 \vec{n}, m^{2}=4, n^{2}=1$ and $\vec{m} \perp \vec{n}$.
11. Given the vertices $A(0,0,0), B(2,-3,5), C(-3,4,-2)$ of parallelogram $A B C D$. Find the coordinates of vertex $D$.

## Answers

1. $\pm \frac{1}{\sqrt{3}}(-2 \vec{i}+2 \vec{j}+\vec{k})$. 2. $\vec{c}=\vec{i}+\vec{k}$ or $c=\frac{1}{3}(-\vec{i}+4 \vec{j}+\vec{k})$. 3. $m=1$. 4. -96. 5. 1) $76^{\circ} 22^{\prime}, 76^{\circ} 22^{\prime}, 27^{\circ} 16^{\prime}$; 2) $3,3, \sqrt{2}$; 3) $49^{\circ} 40^{\prime}$. 6. 7. 7. $90^{\circ}$. 8. $143.10 .113 .11 . D(-5,7,-7)$.

## § 3. Vector Product of Vectors

$1^{\circ}$. Vectors $\vec{a}, \vec{b}$, and $\vec{c}$, brought to a common origin, form a right (left) triplet if: when looking from the end of vector $\vec{c}$ at the plane of vectors $\vec{a}$ and $\vec{b}$, the shortest rotation from $\vec{a}$ to $\vec{b}$ is counterclockwise (clockwise) (Fig. 3.10).

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-038.jpg?height=235&width=258&top_left_y=1097&top_left_x=1491)

Fig. 3.10

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-038.jpg?height=238&width=369&top_left_y=1115&top_left_x=2036)

Fig. 3.11 $2^{\circ}$. The vector product of non-zero vectors $\vec{a}$ and $\vec{b}$ is a vector $\vec{c}$, denoted by $\vec{a} \times \vec{b}$, satisfying the following three conditions.

1) $\vec{a} \times \vec{b} \perp \vec{a}$ and $\vec{a} \times \vec{b} \perp \vec{b}$, i.e., the vector $\vec{a} \times \vec{b}$ is perpendicular to the plane of vectors $\vec{a}$ and $\vec{b}$.
2) The vector $\vec{a} \times \vec{b}$ is directed such that the vectors $\vec{a}, \vec{b}$, and $\vec{a} \times \vec{b}$ form a right triplet.
3) $|\vec{a} \times \vec{b}|=|\vec{a}| \cdot|\vec{b}| \sin \varphi$, i.e., its length is numerically equal to the area of the parallelogram constructed on vectors $\vec{a}$ and $\vec{b}$ (Fig. 3.11), thus, $|\vec{a} \times \vec{b}|=S$.

If vectors $\vec{a}$ and $\vec{b}$ are collinear, then $\vec{a} \times \vec{b}$ is understood to be the zero vector: $\vec{a} \times \vec{b}=\overrightarrow{0} \quad(\sin \varphi=0)$.
$3^{\circ}$. If the coordinates of the vectors being multiplied $\vec{a}=$ $=\left\{a_{x}, a_{y}, a_{z}\right\}, \vec{b}=\left\{b_{x}, b_{y}, b_{z}\right\}$ are known, then the formula for finding the coordinates of the vector product is

$$
\vec{c}=\vec{a} \times \vec{b}=\left|\begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k} \\
a_{x} & a_{y} & a_{z} \\
b_{x} & b_{y} & b_{z}
\end{array}\right|
$$

in which the determinant should be expanded along the elements of the first row.

## Examples with Solutions