Math Concept Check Chapter 4

What is the connection between cartesian geometry and vector geometry? Given a point P in 3-space we associate three numbers x, y, and z with P.  These numbers are called the coordinates of P, and we denote the point (x,y,z) or P(x,y,z) to emphasize the label P.  The result is called a Cartesian Coordinate System for 3-space, and the resulting description in 3-space is called Cartesian geometry.  Vectors are introduced by identifying each point P(x,y,z) with the vector v->=                  in R^3, represented by the arrow from the origin to P.  We say informally that point P has vector v-> and vector v-> has point P.  In particular, the terms vector and point are interchangeable here.  The resulting description of 3-space is called vector geometry where the origin is 0->=                                                                              What is the parallelogram law? What is the scalar multiple law?  In the parallelogram determined by two vectors v-> and w->, the vector v->+w-> is the diagonal with the same tail/starting point as v-> and w->.  If “a” is a real number and v->/=0-> is a vector then,                                                                1. The length of av-> is llav->ll = lal llv->ll    2. If av->/=0->, the direction of av-> is – the same as v-> if a>0 – opposite to v-> if a<0.   What does it mean to say that 2 nonzero vectors are parallel?  How can we tell whether or not two given vectors are parallel?  Two nonzero vectors are called parallel if they have the same or opposite direction.  Two nonzero vectors v-> and w-> are parallel if and only if one is a scalar multiple of the other.  That is, you can multiply the same scalar “a” by all entries in v-> to get w-> and vice versa for another scalar b, where a and b can be any real number.    What is the vector equation of a line?  Show why this definition makes sense.  The line parallel to d->/=0-> through the point with vector Po-> is given by P->=Po-> + td->; t is any scalar.  In other words, the point P-> is on this line if and only if a real number t exists such that P->=Po->+ td->.  This makes sense since: d->/=0-> is a direction vector for a line if it is parallel to AB-> for some pair of distinct points A and B.  We use the fact that there is exactly one line that passes through a particular point Po(xo, yo, zo) and has a given direction vector d->=             .  We want to describe this line by giving a condition on x, y, and z that the point P(x,y,z) lies on this line.  Let Po->=                  and P->=                 denote the vectors of Po-> and P-> respectively.  Then P->=Po->+PoP->.  Hence P lies on the line if and only if PoP->is parallel to d-> – that is, if and only ifPoP->=td-> for some                     scalar t.  Thus P-> is the vector of a point on the line if and only if P->=Po-> + td-> for some scalar t.                                                                    What does it mean to say that two vectors are orthogonal?  How can we tell whether or not two vectors have this property?  Two vectors v-> and w-> are said to be orthogonal if v->=0-> or w->=0-> or the angle between them is pi/2.  Two vectors v-> and w-> are orthogonal if and only if v-> dot product w->=0->.   What is the projection of one vector on another?  How is a projection computed?  When a vector v-> is resolved into two perpendicular components v1-> + v2->, if v2-> is parallel to vector w->, then v2-> is called the projection of v-> on d->.  It is denoted v2->=projd->v->.  The projection of v-> on d-> is given by projd->v->=[(v-> dot product d->)/lld->ll^2]d->.  What is a normal vector for a plane?  How can such a vector be used to obtain an equation for a plane?  A nonzero vector n-> is called a normal for a plane if it is orthogonal to every vector in the plane.  The plane through Po(xo,yo,zo) with normal n->=                       /=0-> as a normal vector is given by a(x-xo)+b(y-yo)+c(z-zo) = 0.  In other words, a point P(x,y,z) is on this plane if and only if x,y, and z satisfy this equation.   How do we define the cross product of two vectors?  How can we use a determinant to compute a cross product?  Given vectors v1->=                                          v2->=                       , define the cross product v1-> X v2-> by  v1-> X v2-> =                                        .  When writing the cross product v1-> X v2-> as a determinant, v1-> X v2-> = det                                                       = l                              l i->  –  l                                             l j-> +l                               l k->                                                                                     where the determinant is expanded along the first                                                    column.  What is the Lagrange Identity?  How can we use it to show that                                                                                                                                                                                                                                                                                           ll u-> X v-> ll = ll u-> ll ll v-> ll sin theta?  If u-> and v-> are any two vectors in R^3, then  llu-> X v-> ll^2  = llu->ll^2 llv->ll^2 – (u-> dot product v->)^2.  Given u-> and v->, introduce a coordinate system and write u-> =                                and v-> =                                      in component form.  An expression for the magnitude of the vector u-> X v-> can be easily obtained from the Lagrange Identity.  If theta is the angle between u-> and v->, substituting  u-> dot product v-> = llu->ll llv->ll cos theta into the Lagrange Identity gives                                                                                                       llu-> X v->ll^2  = llu->ll^2 llv->ll^2  – llu->ll^2 llv->ll^2 cos^2 theta = llu->ll^2 llv->ll^2 sin^2 theta.  But sin theta is nonnegative on the range 0<= theta <= pi, so taking the positive square root of both sides gives llu-> X v->ll = llu->ll llv->ll sin theta.  What does theorem 4.3.4 say, and why does it make sense?  If u-> and v-> are two nonzero vectors and theta is the angle between u-> and v->, then   1. llu-> X v->ll = llu->ll llv->ll sin theta = the area of the parallelogram determined by u-> and v->.  2. u-> and v-> are parallel if and only if u-> X v-> = 0->.  

The parallelogram determined by the vectors u-> and v-> has base length llv->ll and altitude llu->llsin theta.  Hence the area of the parallelogram form by u-> and v-> is (llu->ll sin theta ) llv->ll = llu-> X v-> ll.  By this, u-> X v-> = 0-> if and only if the area of the parallelogram is zero.  The area vanishes if and only if u-> and v-> have the same or opposite direction  – that is, if and only if they are parallel.  What is a parallelopiped, and how can we compute its volume?  If three vectors w->, u->,  and v-> are given, they determine a “squashed” rectangular solid called a parallelepiped, and it is often useful to be able to find the volume of such a solid.  The volume of the parallelepiped determined by three vectors w->, u->,  and v-> is given by l w-> dot product ( u-> X v-> )l

When doing vector dot product vector cross product vector, you can put them in a matrix with the columns as each vector in order in the problem and find the determinant of that.