Hosam Hajjir
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  • Ottawa, ON, Canada
Pairwise-intersecting circles $R,G,B$ have concurrent common chords?
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5 votes

The equations of the three circles are $ (x - x_1)^2 + (y - y_1)^2 = r_1^2 \hspace{30pt}(1) $ $ (x - x_2)^2 + (y - y_2)^2 = r_2^2 \hspace{30pt}(2)$ $ (x - x_3)^2 + (y - y_3)^2 = r_3^2 \hspace{30pt}(3)$...

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Algorithm to determine if a 3D ellipsoid is contained within another?
5 votes

Since the two ellipsoid share the same center, then we can take this center to be the origin of the coordinate system, and then the equations of the two ellipsoids will be $ r^T Q_1 r = 1 $ and $ r^T ...

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Determine the maximum of $f(x) = x + \sqrt{4-x^2}$ without calculus
5 votes

Let $x = 2 \cos \theta$ Then $x + \sqrt{4 - x^2} = 2 \cos \theta + 2 \sin \theta = 2 (\cos \theta + \sin \theta)$ Now, the trigonometric function $ a \cos \theta + b \sin \theta $ satisfies: $$-\sqrt{...

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Finding smallest vector satisfying an equation with a dot product
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5 votes

$ \theta \cdot x = | \theta | | x | \cos \psi = \dfrac{1}{y}$ where $\psi$ is the angle between the vector $x$ and the vector $\theta$. Since $x$ and $y$ are fixed, this implies the minimum $| \theta |...

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Rotating a cone
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4 votes

The equation of a circular cone is determined by two things: It's axis represented by the unit vector $\mathbf{a}$, and the semi-vertical angle which is the angle $\theta$ ​between the axis and the ...

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If $a^{2}+b^{2} \leq 4$, prove that $a+b \leq 4$
4 votes

From Cauchy-Schwartz inequality, $a^2 + b^2 = (a^2 + b^2) ( \dfrac{1}{2} + \dfrac{1}{2} ) \ge (\dfrac{1}{\sqrt{2}} a + \dfrac{1}{\sqrt{2}} b )^2 = \dfrac{1}{2} (a + b)^2$ Hence $(a + b)^2 \le 2 (a^2 + ...

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How to solve $3\sec(x)-2\cot(x)>0$ (trigonometric inequality)?
3 votes

$ 3 \sec x - 2 \cot x \gt 0 $ Implies $ \dfrac{3}{\cos x} \gt 2 \dfrac{\cos x }{\sin x} $ Multiply through by $\sin^2 x \cos^2 x $ $ 3 \cos x \sin^2 x \gt 2 \sin x \cos^3 x $ Hence, $ \cos x \sin x ( ...

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Prove that from $0 > a > b$ follows $0 > b^{-1}>a^{-1}$
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3 votes

Since $0 \gt a \gt b $ both $a $ and $b$ are negative, so their product is positive. Divide the inequality by $ab$, the inequality sign remains the same. So $ \dfrac{0}{ab} \gt \dfrac{a}{ab} \gt \...

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If $[a]_\times$ is a matrix such that $[a]_\times b$ = $a \times b$, and $R$ is a (rotation) Matrix, how to simplify $[Ra]_\times$?
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3 votes

If you rotate both $a$ and $b$ by the matrix $R$ then it is evident (without proof) that $ (R a) \times (R b) = R (a \times b ) $ The left hand side $[ R a ]_\times (R b) $ and the right hand side is $...

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Determining whether a set of points is contained on an ellipse
3 votes

Let $n$ be the number of points. If $n\le 4$ and the points are not collinear, then you can always find an ellipse on which the points lie. If $n \ge 5$, then take any $5$ of the points, and construct ...

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How do we know the position of fixed point in this Q?
2 votes

The equation of the line is $ P(t) = P_0 + t v $ $OP(t) \times v = ((P_0 - O) + t v ) \times v = (P_0 - O) \times v + t (v \times v) = (P_0 - O) \times v $ Thus the cross product is independent of $t$....

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Find a point on a line that creates a perpendicular in 3D space
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2 votes

Since point $D$ will be on $AC$, then there exists a scalar $t \in \mathbb{R} $ such that $ D = A + t (C - A) \hspace{20pt}(1)$ The vector $(C-A)$ is the direction vector of the ray $AC$. Now we want ...

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Find the volume of the solid obtained by rotating $R$ about $y=\frac{x}{5}$.
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2 votes

First find the intersection of the line $y = x$ with the circle. Plug in $y = x$ you get $ (x - 3)^2 + ( x - 4)^2 = 4 $ This simplifies to, $ 2 x^2 - 14 x + 21 = 0 $ By factoring, this becomes, $ 2 (...

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Write a triangle in the space in parametric form
2 votes

Any point $P$ on the perimeter or the interior of $\triangle ABC$ can be written as a linear combination of the coordinates of $A,B,C$ (Barycentric coordinates) $P = t A + s B + (1 - t - s) C $ where ...

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How can I rotate $y = Kx +M$ in 90 degrees in the center of the line - not origin?
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2 votes

If you're rotating by $90^\circ$ counter clockwise about the origin, then the image of the point $(x, y)$ on the line becomes: $(x', y') = (-y, x) $ Thus $x = y'$ and $y = -x'$. Plug these into the ...

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Tetrahedron triangular section
2 votes

The point $P = a A + b B + c C + d D$ where $a + b + c + d=1$ You want to find the intersection of the plane $PCD$ with the segment $AB$. Points in plane $PCD$ are spanned by two vectors: $CD$ and $CP$...

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Condition for line intersection in $\mathbb{R}^3$
2 votes

The actual condition is $ (\mathbf{a_2 } - \mathbf{a_1} ) \cdot (\mathbf{b_2} \times \mathbf{b_1} ) = 0 $ But since for any three vectors $\mathbf{a,b,c}$, $ \mathbf{a}\cdot (\mathbf{b} \times \mathbf{...

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Construct focii of ellipse given center and four tangent lines
2 votes

The equation of an ellipse in the plane is given by $ (r - E)^T Q (r - E) = 1 \hspace{20pt} (1)$ where $E$ is the center, and $Q$ is a $2 \times 2$ symmetric matrix. At any point on the ellipse the ...

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Find a point $P_2$ on an ellipse, whose chord with $P_1$ is a max distance $d$ from its nearest side
2 votes

The parametric equation of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 $ is $ r = (x, y) = ( a \cos t , b \sin t ) \hspace{20pt} t \in \mathbb{R} $ Now $P_1 = (x_1, y_1) = (a \cos t_1, b \sin ...

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Rotation of a non-square matrix
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2 votes

I found an excellent paper that addresses this problem in the $3D$ case, but the methods developed there are applicable in a straight forward manner to the $2D$ case presented in this question. We ...

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Finding the area of a circle tangentially inside a triangle
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2 votes

$AT = AP$ so $AT = 2$ Taking $B$ to be the origin of the Cartesian coordinate system, then $A = (0, 3)$ The unit direction vector from A down the hypotenuse is $(\cos C, - \sin C) = (4/5, -3/5) $ $T = ...

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How can I find the points at which two 2D lines are a specific distance apart?
2 votes

Let the two lines be $\mathbf{P_1}(t) $ and $\mathbf{P_2}(t) $ then we can write $\mathbf{P_1}(t) = \mathbf{Q_1} + t \mathbf{V_1} $ $\mathbf{P_2} (t) = \mathbf{Q_2} + t \mathbf{V_2} $ It can be ...

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Finding the vectorial expression for the mutual slant of two cones with a common vertex
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2 votes

Start by parameterizing $\mathbf{k_2}$ in terms of $\mathbf{k_1}$. Let $\mathbf{u_1}, \mathbf{u_2}$ be two mutually orthogonal unit vectors that are also orthogonal to $\mathbf{k_1}$, then any vector ...

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Find the control point of a quadratic bezier curve with known maximum curve value
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2 votes

$p(t) = (1 - t)^2 P_0 + 2 t (1 - t) P_1 + t^2 P_2 $ Taking the derivative with respect to $t$ gives us the tangent vector $p'(t) = - 2 (1 - t) P_0 + 2 (1 - 2 t) P_1 + 2 t P_2 $ At maximum $x$, the $x$...

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Rotated coordinates on a unit sphere
2 votes

Here is a brute force method to find the new coordinates of $p_3$. Assuming that $\phi$ is the azimuthal angle measured counter clockwise from the positive $x$ axis, and $\theta$ is the polar angle ...

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Shadow time for a satellite on an inclined orbit
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2 votes

Let the radius of Earth be $R_e$, and let the radius of the orbit of the satellite be $R \gt R_e$. Choose the Cartesian frame to have its origin at the center of Earth, and with $x$ axis pointing ...

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How to find the smallest sphere that intersects 3 given lines in 3D?
2 votes

Let the three lines be given by parametric equations $ R_i(t) = P_{i} + t V_{i} ,\hspace{20pt} i = 1, 2, 3 $ where the direction vectors $\{ V_i \}$ are assumed to be unit vectors. Given the distance $...

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Prove vectors with specific length and dot product cannot exist
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2 votes

$ u \cdot v = \|u\| \|v\| \cos \theta $ So $ 3 = (1)(2) \cos \theta $ which implies $\cos \theta = \dfrac{3}{2} $ which is not possible.

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Calculate the height of the cone from the frustum’s bottom and top radius and frustum height.
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2 votes

The figure above summarizes the situation. We have two similar triangles, the big one with height $h$ and base $9$, and the small one with height $(h-8)$ and base $5$. Therefore, by equal ratios of ...

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How to prove $(a_1+a_2+a_3+...+a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+...+\frac{1}{a_n}\right)≥n^2$
2 votes

Let $v = [ \sqrt{a_1} , \sqrt{a_2} , ...., \sqrt{a_n} ]^T $ and $w = [ \dfrac{1}{\sqrt{a_1}}, \dfrac{1}{\sqrt{a_2}}, ..., \dfrac{1}{\sqrt{a_n}}]^T$ The by Cauchy-Schwartz inequality, $ v \cdot w \le | ...

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