Calculus Volume 3

# Key Concepts

Calculus Volume 3Key Concepts

### 2.1Vectors in the Plane

• Vectors are used to represent quantities that have both magnitude and direction.
• We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.
• Subtraction of vectors is defined in terms of adding the negative of the vector.
• A vector is written in component form as $v=〈x,y〉.v=〈x,y〉.$
• The magnitude of a vector is a scalar: $‖v‖=x2+y2.‖v‖=x2+y2.$
• A unit vector $uu$ has magnitude $11$ and can be found by dividing a vector by its magnitude: $u=1‖v‖v.u=1‖v‖v.$ The standard unit vectors are $i=〈1,0〉andj=〈0,1〉.i=〈1,0〉andj=〈0,1〉.$ A vector $v=〈x,y〉v=〈x,y〉$ can be expressed in terms of the standard unit vectors as $v=xi+yj.v=xi+yj.$
• Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.

### 2.2Vectors in Three Dimensions

• The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin. Ordered triples $(x,y,z)(x,y,z)$ are used to describe the location of a point in space.
• The distance $dd$ between points $(x1,y1,z1)(x1,y1,z1)$ and $(x2,y2,z2)(x2,y2,z2)$ is given by the formula
$d=(x2−x1)2+(y2−y1)2+(z2−z1)2.d=(x2−x1)2+(y2−y1)2+(z2−z1)2.$
• In three dimensions, the equations $x=a,y=b,andz=cx=a,y=b,andz=c$ describe planes that are parallel to the coordinate planes.
• The standard equation of a sphere with center $(a,b,c)(a,b,c)$ and radius $rr$ is
$(x−a)2+(y−b)2+(z−c)2=r2.(x−a)2+(y−b)2+(z−c)2=r2.$
• In three dimensions, as in two, vectors are commonly expressed in component form, $v=〈x,y,z〉,v=〈x,y,z〉,$ or in terms of the standard unit vectors, $xi+yj+zk.xi+yj+zk.$
• Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let $v=〈x1,y1,z1〉v=〈x1,y1,z1〉$ and $w=〈x2,y2,z2〉w=〈x2,y2,z2〉$ be vectors, and let $kk$ be a scalar.
• Scalar multiplication: $kv=〈kx1,ky1,kz1〉kv=〈kx1,ky1,kz1〉$
• Vector addition: $v+w=〈x1,y1,z1〉+〈x2,y2,z2〉=〈x1+x2,y1+y2,z1+z2〉v+w=〈x1,y1,z1〉+〈x2,y2,z2〉=〈x1+x2,y1+y2,z1+z2〉$
• Vector subtraction: $v−w=〈x1,y1,z1〉−〈x2,y2,z2〉=〈x1−x2,y1−y2,z1−z2〉v−w=〈x1,y1,z1〉−〈x2,y2,z2〉=〈x1−x2,y1−y2,z1−z2〉$
• Vector magnitude: $‖v‖=x12+y12+z12‖v‖=x12+y12+z12$
• Unit vector in the direction of v: $v‖v‖=1‖v‖〈x1,y1,z1〉=〈x1‖v‖,y1‖v‖,z1‖v‖〉,v‖v‖=1‖v‖〈x1,y1,z1〉=〈x1‖v‖,y1‖v‖,z1‖v‖〉,$ $v≠0v≠0$

### 2.3The Dot Product

• The dot product, or scalar product, of two vectors $u=〈u1,u2,u3〉u=〈u1,u2,u3〉$ and $v=〈v1,v2,v3〉v=〈v1,v2,v3〉$ is $u·v=u1v1+u2v2+u3v3.u·v=u1v1+u2v2+u3v3.$
• The dot product satisfies the following properties:
• $u·v=v·uu·v=v·u$
• $u·(v+w)=u·v+u·wu·(v+w)=u·v+u·w$
• $c(u·v)=(cu)·v=u·(cv)c(u·v)=(cu)·v=u·(cv)$
• $v·v=‖v‖2v·v=‖v‖2$
• The dot product of two vectors can be expressed, alternatively, as $u·v=‖u‖‖v‖cosθ.u·v=‖u‖‖v‖cosθ.$ This form of the dot product is useful for finding the measure of the angle formed by two vectors.
• Vectors u and v are orthogonal if $u·v=0.u·v=0.$
• The angles formed by a nonzero vector and the coordinate axes are called the direction angles for the vector. The cosines of these angles are known as the direction cosines.
• The vector projection of v onto u is the vector $projuv=u·v‖u‖2u.projuv=u·v‖u‖2u.$ The magnitude of this vector is known as the scalar projection of v onto u, given by $compuv=u·v‖u‖.compuv=u·v‖u‖.$
• Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector F and the displacement is represented by the vector s, then the work done W is given by the formula $W=F·s=‖F‖‖s‖cosθ.W=F·s=‖F‖‖s‖cosθ.$

### 2.4The Cross Product

• The cross product $u×vu×v$ of two vectors $u=〈u1,u2,u3〉u=〈u1,u2,u3〉$ and $v=〈v1,v2,v3〉v=〈v1,v2,v3〉$ is a vector orthogonal to both $uu$ and $v.v.$ Its length is given by $‖u×v‖=‖u‖·‖v‖·sinθ,‖u×v‖=‖u‖·‖v‖·sinθ,$ where $θθ$ is the angle between $uu$ and $v.v.$ Its direction is given by the right-hand rule.
• The algebraic formula for calculating the cross product of two vectors,
$u=〈u1,u2,u3〉andv=〈v1,v2,v3〉,u=〈u1,u2,u3〉andv=〈v1,v2,v3〉,$ is
$u×v=(u2v3−u3v2)i−(u1v3−u3v1)j+(u1v2−u2v1)k.u×v=(u2v3−u3v2)i−(u1v3−u3v1)j+(u1v2−u2v1)k.$
• The cross product satisfies the following properties for vectors $u,v,andw,u,v,andw,$ and scalar $c:c:$
• $u×v=−(v×u)u×v=−(v×u)$
• $u×(v+w)=u×v+u×wu×(v+w)=u×v+u×w$
• $c(u×v)=(cu)×v=u×(cv)c(u×v)=(cu)×v=u×(cv)$
• $u×0=0×u=0u×0=0×u=0$
• $v×v=0v×v=0$
• $u·(v×w)=(u×v)·wu·(v×w)=(u×v)·w$
• The cross product of vectors $u=〈u1,u2,u3〉u=〈u1,u2,u3〉$ and $v=〈v1,v2,v3〉v=〈v1,v2,v3〉$ is the determinant $|ijku1u2u3v1v2v3|.|ijku1u2u3v1v2v3|.$
• If vectors $uu$ and $vv$ form adjacent sides of a parallelogram, then the area of the parallelogram is given by $‖u×v‖.‖u×v‖.$
• The triple scalar product of vectors $u,u,$ $v,v,$ and $ww$ is $u·(v×w).u·(v×w).$
• The volume of a parallelepiped with adjacent edges given by vectors $u,v,andwu,v,andw$ is $V=|u·(v×w)|.V=|u·(v×w)|.$
• If the triple scalar product of vectors $u,v,andwu,v,andw$ is zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.
• The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.
• Torque $ττ$ measures the tendency of a force to produce rotation about an axis of rotation. If force $FF$ is acting at a distance $rr$ from the axis, then torque is equal to the cross product of $rr$ and $F:F:$ $τ=r×F.τ=r×F.$

### 2.5Equations of Lines and Planes in Space

• In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector $v=〈a,b,c〉v=〈a,b,c〉$ passing through point $P=(x0,y0,z0)P=(x0,y0,z0)$ is $r=r0+tv,r=r0+tv,$ where $r0=〈x0,y0,z0〉r0=〈x0,y0,z0〉$ is the position vector of point $P.P.$ This equation can be rewritten to form the parametric equations of the line: $x=x0+ta,x=x0+ta,$ $y=y0+tb,y=y0+tb,$ and $z=z0+tc.z=z0+tc.$ The line can also be described with the symmetric equations $x−x0a=y−y0b=z−z0c.x−x0a=y−y0b=z−z0c.$
• Let $LL$ be a line in space passing through point $PP$ with direction vector $v.v.$ If $QQ$ is any point not on $L,L,$ then the distance from $QQ$ to $LL$ is $d=‖PQ→×v‖‖v‖.d=‖PQ→×v‖‖v‖.$
• In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
• Given a point $PP$ and vector $n,n,$ the set of all points $QQ$ satisfying equation $n·PQ→=0n·PQ→=0$ forms a plane. Equation $n·PQ→=0n·PQ→=0$ is known as the vector equation of a plane.
• The scalar equation of a plane containing point $P=(x0,y0,z0)P=(x0,y0,z0)$ with normal vector $n=〈a,b,c〉n=〈a,b,c〉$ is $a(x−x0)+b(y−y0)+c(z−z0)=0.a(x−x0)+b(y−y0)+c(z−z0)=0.$ This equation can be expressed as $ax+by+cz+d=0,ax+by+cz+d=0,$ where $d=−ax0−by0−cz0.d=−ax0−by0−cz0.$ This form of the equation is sometimes called the general form of the equation of a plane.
• Suppose a plane with normal vector n passes through point $Q.Q.$ The distance $DD$ from the plane to point $PP$ not in the plane is given by
$D=‖projnQP→‖=|compnQP→|=|QP→·n|‖n‖.D=‖projnQP→‖=|compnQP→|=|QP→·n|‖n‖.$
• The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
• The measure of the angle $θθ$ between two intersecting planes can be found using the equation: $cosθ=|n1·n2|‖n1‖‖n2‖,cosθ=|n1·n2|‖n1‖‖n2‖,$ where $n1n1$ and $n2n2$ are normal vectors to the planes.
• The distance $DD$ from point $(x0,y0,z0)(x0,y0,z0)$ to plane $ax+by+cz+d=0ax+by+cz+d=0$ is given by
$D=|a(x0−x1)+b(y0−y1)+c(z0−z1)|a2+b2+c2=|ax0+by0+cz0+d|a2+b2+c2.D=|a(x0−x1)+b(y0−y1)+c(z0−z1)|a2+b2+c2=|ax0+by0+cz0+d|a2+b2+c2.$

• A set of lines parallel to a given line passing through a given curve is called a cylinder, or a cylindrical surface. The parallel lines are called rulings.
• The intersection of a three-dimensional surface and a plane is called a trace. To find the trace in the xy-, yz-, or xz-planes, set $z=0,x=0,ory=0,z=0,x=0,ory=0,$ respectively.
• Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form $Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.$
• To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.
• Important quadric surfaces are summarized in Figure 2.87 and Figure 2.88.

### 2.7Cylindrical and Spherical Coordinates

• In the cylindrical coordinate system, a point in space is represented by the ordered triple $(r,θ,z),(r,θ,z),$ where $(r,θ)(r,θ)$ represents the polar coordinates of the point’s projection in the xy-plane and $zz$ represents the point’s projection onto the z-axis.
• To convert a point from cylindrical coordinates to Cartesian coordinates, use equations $x=rcosθ,x=rcosθ,$ $y=rsinθ,y=rsinθ,$ and $z=z.z=z.$
• To convert a point from Cartesian coordinates to cylindrical coordinates, use equations $r2=x2+y2,r2=x2+y2,$ $tanθ=yx,tanθ=yx,$ and $z=z.z=z.$
• In the spherical coordinate system, a point $PP$ in space is represented by the ordered triple $(ρ,θ,φ),(ρ,θ,φ),$ where $ρρ$ is the distance between $PP$ and the origin $(ρ≠0),(ρ≠0),$ $θθ$ is the same angle used to describe the location in cylindrical coordinates, and $φφ$ is the angle formed by the positive z-axis and line segment $OP—,OP—,$ where $OO$ is the origin and $0≤φ≤π.0≤φ≤π.$
• To convert a point from spherical coordinates to Cartesian coordinates, use equations $x=ρsinφcosθ,x=ρsinφcosθ,$ $y=ρsinφsinθ,y=ρsinφsinθ,$ and $z=ρcosφ.z=ρcosφ.$
• To convert a point from Cartesian coordinates to spherical coordinates, use equations $ρ2=x2+y2+z2,ρ2=x2+y2+z2,$ $tanθ=yx,tanθ=yx,$ and $φ=arccos(zx2+y2+z2).φ=arccos(zx2+y2+z2).$
• To convert a point from spherical coordinates to cylindrical coordinates, use equations $r=ρsinφ,r=ρsinφ,$ $θ=θ,θ=θ,$ and $z=ρcosφ.z=ρcosφ.$
• To convert a point from cylindrical coordinates to spherical coordinates, use equations $ρ=r2+z2,ρ=r2+z2,$ $θ=θ,θ=θ,$ and $φ=arccos(zr2+z2).φ=arccos(zr2+z2).$