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Key Concepts

Key Concepts

2.1 Vectors 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.

2.2 Vectors 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

2.3 The 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 uu and vv 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 vv onto uu 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 vv onto uu, 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.

2.4 The 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.

2.5 Equations 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.

2.6 Quadric Surfaces

  • 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.

2.7 Cylindrical 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).