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Home Question Bank Math Geometry Archive 2025 January 13

Geometry Questions from Jan 13,2025

Browse the Geometry Q&A Archive for Jan 13,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Find the equation of a plane through the points \( (2,1,2),(6,-6,-2) ;(4,6,3) \). Question Help: \( \square \) Video Find the equation of the plane containing the point \( (-1,-4,-8) \) and parallel to the plane \( 7 x-6 y-7 z=0 \). Find the equation of a plane through the point \( (6,3,7) \) and orthogonal to the line \( \left\{\begin{array}{l}x(t)=6+8 t \\ y(t)=-3-7 t \\ z(t)=-1+6 t\end{array}\right. \) Find the distance between the skew lines \( \vec{r}(t)=\langle-1,4,-5\rangle t+\langle 6,0,-6\rangle \) anc \( \vec{p}(s)=\langle 0,2,5\rangle s+\langle-4,-3,-7\rangle \). Find the distance from the point \( (3,-6,-3) \) to the line \( \vec{a}(t)=\langle-2,-5,-4\rangle t+\langle 5,6,1\rangle \) Find the (perpendicular) distance from the line given by the parametric equations \[ \begin{array}{l}x(t)=-3-2 t \\ y(t)=3+4 t \\ z(t)=-1-10 t\end{array} \] to the point \( (8,6,-2) \) Let \( \vec{a}(t)=-4,4,-3\rangle t+\langle 1,2,-1\rangle \) and \( \vec{b}(u)=\langle 2,-3,-1\rangle u+\langle-3,8,-4\rangle \). Find the acute angle (in degrees) between the lines: Convert the line determined by the symmetric equations \( \frac{x-5}{10}=\frac{y}{7}=\frac{z+2}{-3} \) to parametric form. \( x(t)=\square \) \( y(t)=\square \) \( z(t)=\square \) 1 Find a vector parallel to the line defined by the parametric equations \( \left\{\begin{array}{l}x(t)=1-9 t \\ y(t)=-1-5 t \\ z(t)=2+6 t\end{array}\right. \) Additionally, find a point on the line. Parallel vector (in angle bracket notation): Find the area of a parallelogram formed by the vectors \( P Q \) and \( P R \), where \( P=(0,-1,0) \), \( Q=(1,1,2) \) and \( R=(-2,1,5) \). ,
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