÷ÈÓ°Ö±²¥

Explain why two intersecting lines determine a unique plane. Explain how you would use the equations of the lines to find the equation of the plane.

Explain why two distinct parallel lines determine a unique plane. Explain how you would use the equations of the lines to find the equation of the plane.

If $||u×v||=||u||||v||$, what is the geometric relationship between u and v?

What point is symmetric to the point $(3,−7,−4)$with respect to the plane $z=1?$

Explain why two nonparallel vectors and a point uniquely determine a plane containing both vectors and the point.

Find an equation of the line containing the given point and parallel to the given vector. Express your answer

(a) as a vector parametrization

(b) in terms of parametric equations

(c) in symmetric form.

$P(0,0,0),d=\left(1,2,−4\right)$

Find $u+vandu-v$also sketch $u,v,u+v,u-v$

$u=\left(4,0\right)andv=\left(0,3\right)$

Find the smallest value of \(n\) so that \(R_n\le10^{-6}\).

\(\sum_{k=1}^{\infty}\frac{k}{e^k}\)

Find a unit vector in the direction opposite to.

A sphere is inscribed in a cube so that each face of the cube is tangent to the sphere. A smaller sphere is inscribed in the cube so that this sphere is tangent to three sides of the cube and is tangent to the larger sphere, as shown in the following figure:

What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? (Hint: Try Exercise 49 first.)