91Ó°ÊÓ

In Exercises 28-31, find ${\mathrm{proj}}_{\mathbf{u}}\mathbf{v}\mathbf{}\mathrm{and}\mathbf{}{\mathrm{proj}}_{\mathbf{v}}\mathbf{u}$.

Find $\stackrel{→}{PQ}$.

$P=(3,6),Q=(−3,−2)$

Find an equation of the line containing the

given pair of points. Express your answer

(a) as a vector parametrization.

(b) in terms of parametric equations.

(c) in symmetric form.

$P(0,0,0),Q(4,−1,6)$

Find the equations of the planes determined by the given conditions.

The plane contains the point $(−4,1,3)$and the line determined

by$\mathbf{r}\left(t\right)=⟨3+7t,-2+t,3+4tâŸ\copyright$

Find the angle between \(u=<2,4,-1>\) and \(v=<0,-3,2>\).

In Exercises 22–29 compute the indicated quantities when $u=(2,1,−3),v=(4,0,1),\mathrm{and}w=(−2,6,5)$

Find the volume of the parallelepiped determined by vectors u, v and w. Do u, v and w form a right-handed triple?

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 1)$$ that trace the unit circle exactly once clockwise, starting at the point $$(1, 0)$$.

(b) Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 2π)$$ that trace the circle centered at $$(2, −3)$$ with radius 5 exactly once counterclockwise, starting at the point $$(7, −3)$$.

(c) Parametric equations $$x = f(t), y = g(t)$$ whose graph is not the graph of a function $$y = f(x)$$.

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading:

(a) A function $f$whose area accumulation function is negative on $[0,5].$

(b) A function $f$whose area accumulation function is decreasing on $[0,5].$

(c) Three antiderivatives of$e^{{\sin }^{2}x}.$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A function that is decreasing on $(-∞,0)$, increasing on $(0,∞)$, and undefined at $x=0$.

(b) A function that is decreasing on $(-∞,0]$and increasing on $[0,∞)$.

(c) A function that is always positive and always decreasing, on all of $R$.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A plane parallel to $x+3y−4z=7$

(b) A line orthogonal to the plane$x+3y−4z=7$

(c) A plane orthogonal to the line $x=3t−5,y=−2t+7,z=−4$