Cross product and sum of vectors

Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be three vectors such that

$$\mathbf{a \times b} = \mathbf{b \times c} =\mathbf{c \times a} \neq \mathbf{0}$$

Prove that $\mathbf{a+b+c}=\mathbf{0}$.

Solution

On an ellipse!

Given the ellipse ${\rm C}: x^2-xy+y^2=\frac{3}{4}$

1. Find its area.
2. Find the points of the ellipse that a horizontal tangent attaches them as well as the points that a vertical tangent attaches them.
3. Which are the most far most and most close points of the ellipse?

Solution

Distance and locus

A line in space passes though the origin and forms equal angles with the axis. If the line intersects the plane $3x+5y+2z=30$ at the point ${\rm P}$ then:

1. Evaluate the distance between ${\rm P}$ and ${\rm O}$ , where ${\rm O}$ is the origin.
2. Find the equations of the locus of the points ${\rm M}$, for which the relation: $$\left | \overrightarrow{{\rm OM}} \right |= \left | \overrightarrow{\rm {PM}} \right |=4$$ holds. 

Solution

Jacobi Identity

Let $\mathbf{a, b, c} \in \mathbb{R}^3$. Prove the following identity:

$$\mathbf{a}\left( \mathbf{b}\times \mathbf{c} \right)+\mathbf{b} \left( \mathbf{c} \times \mathbf{a} \right) + \mathbf{c} \left( \mathbf{a} \times \mathbf{b} \right)=0$$

which is known as Jacobi's identity and give a geometrical interpretation.

Solution

Vector equation

Let $\mathbf{a, b, c} \in \mathbb{R}^3$ be three vectors. Solve the equation:

$$\mathbf{x+(x\cdot a)b = c}$$

Solution

Vectors and outer products

Let $\mathbf{a, \; b}$ be unitary vectors and perpendicular to each other. Let $\mathbf{u}$ be a vector defined as:

$$\mathbf{u}= \left ( \dots \left ( \left ( \left ( \mathbf{a}\times \mathbf{b} \right )\times \mathbf{b} \right )\times \mathbf{b} \right )\dots \right )\times \mathbf{b}$$

where $\mathbf{b}$ is repeated $2011$ times. Prove that:

$$\mathbf{u}=-\mathbf{a \times b}$$

Solution

Length of vector

Let $\vec{a}, \vec{b}$ be vectors such that:

$$|\vec{a}|=11, \;\; |\vec{b}|=23, \;\; |\vec{a}-\vec{b}|=30$$

Find the length of the vector $\vec{a}+\vec{b}$.

Solution

Volume of a solid

Evaluate the volume of the solid:

$$V=\{(x, y, z)\in \mathbb{R}^3: x \in [-1, 1], \;\; y^2+ z^2 \leq x^2$$

Solution

Equation of sphere

Given the circle with center $K(5, 4, 0)$ and radius $r=6$ in the $xy$ plane , find the sphere that passes through that circle and touches the plane $P: 3x+2y+6z=1$.

Solution

Vector calculus

Let $\alpha = {\overrightarrow{\rm OA}}, \; \beta = \overrightarrow{\rm OB}, \; \gamma = \overrightarrow{\rm O\Gamma}$ be unitary vectors that form equal angle $\dfrac{\pi}{3}$, that is:

$$\bigl({\widehat{\vec{\alpha},\vec{\beta}}\,}\bigr)=\bigl({\widehat{\vec{\beta},\vec{\gamma}}\,}\bigr)=\bigl({\widehat{\vec{\gamma},\vec{\alpha}}\,}\bigr)=\dfrac{\pi}{3}$$

Evaluate the value of

$$A=\bigl({\overrightarrow{\alpha}\times\bigl({\overrightarrow{\alpha}\times\bigl({\overrightarrow{\alpha}\times\overrightarrow{\gamma}}\bigr)}\bigr)}\bigr)\cdot\bigl({\overrightarrow{\beta}\times\overrightarrow{\gamma}}\bigr)$$

Sphere

Given the sphere $\mathbb{S}: x^2 +y^2 +z^2 -2x + 2y -4z +2 =0$ and the plane $(\pi): 3x-2y +z =5$ show that:

a) the two surfaces intersect.
b) The radius of the circle formed by the intersection is $\displaystyle R = \frac{26}{7}$. In continuity evaluate its center.

Solution

Locus of poles of ellipse

Let $M$ be an arbitrary point of an ellipse $\displaystyle \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 $. We draw the perpendicular lines to its axis $MA, \; MB$ respectively. Prove that the pole of the line passing through the point $A, \; B$ lies on the curve $\displaystyle \frac{a^2}{x^2}+ \frac{b^2}{y^2}=1 $.

Solution:

Volume of tetrahedron

A pyramid has vertices the points $A(2, 0, 0), \; B(0, 3, 0) , \;C (0, 0, 6), \; D(2, 3, 8)$. Evaluate its volume and the lenght of the  height from the vertice $D$.

Solution

Zero vector perpendicular to all others

If the vector $\mathbf{v}$ is perpendicular to three non coplanar vectors, then prove that $\mathbf{v}=0$.

Solution: