Chapter 10: Vectors and the geometry of space

Find the work done by a force \({\bf{F}} = 8{\bf{i}} - 6{\bf{j}} + 9{\bf{k}}\) that moves an object from the point \((0,10,8)\) to the point \((6,12,20)\) along a straight line. The distance is measured in meters and the force in newtons.

To determine

Whether the planes \(x = 4y - 2z\) and \(8y = 1 + 2x + 4z\) are parallel, perpendicular, or neither.

To d\({\bf{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle \)escribe all set of points for condition \(\left| {{\bf{r}} - {{\bf{r}}_0}} \right| = 1\).

37-38= Use the formula in Exercise 36 to find the curvature.s

37. \(x = {e^t}cost,\;\;\;y = {e^t}sint\)

If \({\rm{u}}\)and \({\rm{v}}\)are the vectors shown in figure, find \({\rm{u}} \cdot {\rm{v}}\)and\(\left| {{\rm{u}} \times {\rm{v}}} \right|\). Is \(u \times v\)directed into the page or out of it?

To determine

Whether the points \(A(1,3,2),B(3, - 1,6),C(5,2,0)andD(3,6, - 4)\) lie on the same plane.

To determine

Whether the planes \(x + 2y + 2z = 1\) and \(2x - y + 2z = 1\) are parallel, perpendicular, or neither.

A tow truck drags a stalled car along a road. The chain makes an angle of \({30^circ }\) with the road and the tension in the chain is \(1500\;{\rm{N}}\). How much work is done by the truck in pulling the car \(1\;{\rm{km}}\) ?

(a) Sketch the plane curve with the given vector equation

(b) Find

(c) Sketch the position vectorand the tangent vectorfor the given value of t

To describe the set of all points for condition \(\left| {{\bf{r}} - {{\bf{r}}_1}} \right| + \left| {{\bf{r}} - {{\bf{r}}_2}} \right| = k\).