- 12.1: What is the difference between a vector and a scalar?
- 12.2: How do you add two vectors geometrically? How do you add them algeb...
- 12.3: If a is a vector and c is a scalar, how is ca related to a geo metr...
- 12.4: How do you find the vector from one point to another?
- 12.5: How do you find the dot product of two vectors if you know their le...
- 12.6: How are dot products useful?
- 12.7: Write expressions for the scalar and vector projections of b onto a...
- 12.8: How do you find the cross product a _ b of two vectors if you know ...
- 12.9: How are cross products useful?
- 12.10: (a) How do you find the area of the parallelogram deter mined by a ...
- 12.11: How do you find a vector perpendicular to a plane?
- 12.12: How do you find the angle between two intersecting planes?
- 12.13: Write a vector equation, parametric equations, and sym metric equat...
- 12.14: Write a vector equation and a scalar equation for a plane.
- 12.15: (a) How do you tell if two vectors are parallel? (b) How do you tel...
- 12.16: (a) Describe a method for determining whether three points , , and ...
- 12.17: (a) How do you find the distance from a point to a line? (b) How do...
- 12.18: What are the traces of a surface? How do you find them?
- 12.19: Write equations in standard form of the six types of quadric surfaces.
- 12.20: Determine whether the statement is true or false. If it is true, ex...
- 12.21: Determine whether the statement is true or false. If it is true, ex...
- 12.22: Determine whether the statement is true or false. If it is true, ex...
- 12.23: Determine whether the lines given by the symmetric equations and ar...
- 12.24: (a) Show that the planes and are neither parallel nor perpendicular...
- 12.25: Find an equation of the plane through the line of intersection of t...
- 12.26: (a) Find an equation of the plane that passes through the points , ...
- 12.27: Find the distance between the planes and .
- 12.28: Identify and sketch the graph of each surface.
- 12.29: Identify and sketch the graph of each surface.
- 12.30: Identify and sketch the graph of each surface.
- 12.31: Identify and sketch the graph of each surface.
- 12.32: Identify and sketch the graph of each surface.
- 12.33: Identify and sketch the graph of each surface.
- 12.34: Identify and sketch the graph of each surface.
- 12.35: Identify and sketch the graph of each surface.
- 12.36: Identify and sketch the graph of each surface.
- 12.37: An ellipsoid is created by rotating the ellipse about the -axis. Fi...
Solutions for Chapter 12: VECTORS AND THE GEOMETRY OF SPACE
Full solutions for Multivariable Calculus, | 7th Edition
An equation that relates variable quantities associated with phenomena being studied
An experiment in which subjects do not know if they have been given an active treatment or a placebo
In statistics, a nonnumerical variable such as gender or hair color. Numerical variables like zip codes, in which the numbers have no quantitative significance, are also considered to be categorical.
A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable
p = g(x), where x represents demand and p represents price
An angle formed by two intersecting planes,
Domain of a function
The set of all input values for a function
An identity involving a trigonometric function of 2u
Elements of a matrix
See Matrix element.
Graph of a relation
The set of all points in the coordinate plane corresponding to the ordered pairs of the relation.
Hyperboloid of revolution
A surface generated by rotating a hyperbola about its transverse axis, p. 607.
Midpoint (in a coordinate plane)
For the line segment with endpoints (a,b) and (c,d), (aa + c2 ,b + d2)
tan ?= sin ?cos ?and cot ?= cos ? sin ?
See Viewing window.
A positive number written as c x 10m, where 1 ? c < 10 and m is an integer.
A function that can be written in the form f(x) = a sin (b (x - h)) + k or f(x) = a cos (b(x - h)) + k. The number a is the amplitude, and the number h is the phase shift.
Solution of an equation or inequality
A value of the variable (or values of the variables) for which the equation or inequality is true
symmetric about the x-axis
A graph in which (x, -y) is on the graph whenever (x, y) is; or a graph in which (r, -?) or (-r, ?, -?) is on the graph whenever (r, ?) is
Terminal side of an angle
The x-value of the right side of the viewing window,.