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Find a vector of length 3 that points in the direction opposite to.

In Exercises 54–57 the coordinates of points P, Q, R, and S are given. (a) Show that the four points are coplanar. (b) Determine whether quadrilateral PQRS is a parallelogram. (c) Find the area of quadrilateral PQRS.

P(3, 4, −2), Q(7, 0, 6), R(2, 1, 7), S(5, −2, 13)

What is Lagrange’s identity? How is it used to understand the geometry of the cross product?

Use the Intermediate Value Theorem to prove that every cubic function $f\left(x\right)=A{x}^3+B{x}^2+Cx+D$has at least one real root. You will have to first argue that you can find real numbers a and b so that f(a) is negative and f(b) is positive.