91影视

What is the definition of a sphere?

If u and v are nonzero vectors in ${\mathrm{鈩�}}^3$, why do the equations role="math" localid="1649263352081" $u路(u脳v)=0$ and $v路(u脳v)=0$ tell us that the cross product is orthogonal to both u and v?

In Exercises 3鈥�8, let A, B, C, D, ... , Z be points in $R^3$. Simplify the given quantity.

$\stackrel{鈫�}{AB}+\stackrel{鈫�}{BC}+\stackrel{鈫�}{CA}$

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible:

what it means for a function $f$ to be differentiable, left differentiable, and right differentiable at a point$x=c$

Let u and v be vectors in ${\mathrm{鈩�}}^3$. Prove Lagrange鈥檚 identity, Theorem 10.30:

role="math" localid="1649924794767" ${||u脳v||}^2={||u||}^2{||v||}^2鈭�{(u路v)}^2$

Let u, v and w be three vectors in ${\mathrm{鈩�}}^3$ in which the components of each vector are integers.

(a) Prove that the volume of the parallelepiped determined by u, v and w is an integer.

(b) Find examples of vectors u and v with integer components that show that the area of the parallelogram determined by u and v can be either an integer or an irrational number.

Discuss the boundedness and monotonicity of the

geometric sequence {cr k}鈭�

k=0 with c > 0 and r < 0. In ad-

dition, determine whether the sequence converges or di-

verges. If it converges, find the limit of the sequence.

Let $\mathbf{u},\mathbf{v}$, and $\mathbf{w}$be three mutually perpendicular vectors in ${\mathrm{鈩�}}^3$.

(a) Prove that $\mathbf{u}脳(\mathbf{v}脳\mathbf{w})=\mathbf{0}$.

(b) Show that $|\mathbf{u}路(\mathbf{v}脳\mathbf{w}\left)\right|=鈥�\mathbf{u}鈥�鈥�\mathbf{v}鈥�鈥�\mathbf{w}鈥�$.

Let u and v be vectors in ${\mathrm{鈩�}}^3$ such that $u路v=0$. Prove that if $胃$ is the angle between u and v, then role="math" localid="1650011942678" $\tan 胃=\frac{||u脳v||}{u路v}$

Let u, v, andw be vectors in ${\mathrm{鈩�}}^3$. Prove that $u脳v=u脳w$if and only if u is parallel to $v鈭�w$.