Chapter Review

• The right-hand rule (p. 695)
• The distance formula in space: (p. 696) \(\left\vert P_{1}~P_{2}\right\vert =\sqrt{( x_{2}-x_{1}) ^{2}+( y_{2}-y_{1}) ^{2}+( z_{2}-z_{1}) ^{2}}\)
• The equation of a sphere: (p. 697) \(( x-x_{0}) ^{2}+( y-y_{0}) ^{2}+( z-z_{0}) ^{2}=R^{2}\) Center: \(( x_{0},y_{0},z_{0})\); Radius: \(R\)

• Vectors (p. 699)

Properties of vector addition: (p. 701)

• Commutative property: \(\mathbf{w+v=v}+\mathbf{w}\)
• Associative property: \(\ \ \ \mathbf{u}+\left( \mathbf{v+w}\right) =\left( \mathbf{u}+\mathbf{v}\right) +\mathbf{w}\)
• Additive identity property:\(\ \ \ \mathbf{v+0=0}+\mathbf{v=v}\)
• Additive inverse property: \(\ \ \mathbf{v}+\left( -\mathbf{v}\right) =\left( -\mathbf{v}\right) +\mathbf{v=0}\)

Properties of scalar multiplication: (p. 701)

If \(\mathbf{v}\) and \(\mathbf{w}\) are any two vectors, and \(a\) and \(b\) are scalars, then:

• \(0\mathbf{v}=\mathbf{0}\)
• \(1\mathbf{v=v}\)
• \((-1)\mathbf{v=-v}\)
• \(( a+b) \mathbf{v} = a\mathbf{v}+b\mathbf{v}\)
• \(a(\mathbf{v+w}) =a\mathbf{v}+a\mathbf{w}\)
• \(a( b\mathbf{v}) =( ab) \mathbf{v}\)

• Two vectors are equal if and only if their components are equal. (p. 705)
• Two nonzero vectors \(\mathbf{v}\) and \(\mathbf{w}\) are parallel if there is a nonzero scalar \(a\) for which \(\mathbf{v}=a\,\mathbf{w}\). (p. 706)

  753

• Magnitude of a vector \(\mathbf{v}\) in the plane: (p. 707) \[ \left\Vert \mathbf{v}\right\Vert =\sqrt{v_{1}^{2}+v_{2}^{2}},\quad \hbox{where } \mathbf{v }=v_{1}\mathbf{i}+ v_{2}\mathbf{j} \]

  in space: \[ \left\Vert \mathbf{v}\right\Vert =\sqrt{v_{1}^{2}+v_{2}^{ 2}+v_{3}^{2}},\quad \hbox{where } \mathbf{v}=v_{1}\mathbf{i}+ v_{2}\mathbf{j}+v_{3}\mathbf{k} \]

• A unit vector \(\mathbf{u}\) has magnitude \(1\); that is, \(\Vert \mathbf{u} \Vert =1\). (p. 708)
• For any nonzero vector \(\mathbf{v}\), \(\mathbf{u}=\dfrac{\mathbf{v}}{ \Vert \mathbf{v\Vert }}\) is a unit vector that has the same direction as \( \mathbf{v}\). (p. 708)

• Standard basis vectors in the plane: (pp. 708-709) \[ \mathbf{i} =\langle 1,0\rangle\quad \mathbf{j} =\langle 0,1\rangle \]

  in space: \[ \mathbf{i} =\langle 1,0,0\rangle\quad \mathbf{j} =\langle 0,1,0\rangle\quad \mathbf{k} =\langle 0,0,1\rangle \]

• If \(\mathbf{v}\) is a nonzero vector in the plane, then \(\mathbf{v} =\left\Vert \mathbf{v}\right\Vert ( \cos \theta \mathbf{i}+\sin \theta \mathbf{j})\), where \(\theta\) is the angle between \(\mathbf{ v}\) and the positive \(x\)-axis. (p. 709)

Dot product: (p. 715)

• If \(\mathbf{v}\) \(=v_{1}\mathbf{i}+v_{2}\mathbf{j}\) and \(\mathbf{w}\) \(=w_{1}\mathbf{i}+w_{2}\,\mathbf{j}\), then \(\mathbf{v}\,{\cdot}\, \mathbf{w} =v_{1}w_{1}+v_{2}w_{2}\).
• If \(\mathbf{v}\) \(=v_{1}\mathbf{i}+v_{2}\,\mathbf{j}+v_{3}\mathbf{k}\) and \(\mathbf{w}\) \(=w_{1}\mathbf{i}+w_{2}\,\mathbf{j}+w_{3}\mathbf{k}\), then \( \mathbf{v}\,{\cdot}\, \mathbf{w}=v_{1}w_{1}+v_{2}w_{2}+v_{3}w_{3}.\)

Properties of the dot product: (p. 716)

• Magnitude property: \( \mathbf{v}\,{\cdot}\, \mathbf{v}=\left\Vert \mathbf{v} \right\Vert ^{2}\)
• Commutative property: \( \mathbf{u\,{\cdot}\, v=v\,{\cdot}\, u}\)
• Distributive property: \( \mathbf{u}\,{\cdot}\, (\mathbf{v}+\mathbf{w})= \mathbf{u}\,{\cdot}\, \mathbf{v}+\mathbf{u}\,{\cdot}\, \mathbf{w}\)
• Linearity property: \(a ( \mathbf{u\,{\cdot}\, v} ) =( a\mathbf{u}) \,{\cdot}\, \mathbf{v}\)
• Zero-vector property:  \(\mathbf{0}\,{\cdot}\, \mathbf{v}=0\)
• The angle \(\theta \) between two nonzero vectors \(\mathbf{v}\) and \( \mathbf{w}\): \(\cos \theta =\dfrac{\mathbf{v}\,{\cdot}\, \mathbf{w}}{\left\Vert \mathbf{v}\,\right\Vert \left\Vert \mathbf{w}\right\Vert };\) \(0\leq \theta \leq \pi \) (p. 717)
• Two nonzero vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal if and only if \(\mathbf{v}\,{\cdot}\, \mathbf{w}=0\). (p. 718)
• Direction angles \(\alpha ,\) \(\beta ,\) \(\gamma \) of a nonzero vector \( \mathbf{v}=v_{1}\mathbf{i}+v_{2}\mathbf{j}+v_{3}\mathbf{k}\): (p. 719) \[ \begin{eqnarray*} \cos \alpha &=& \dfrac{v_{1}}{\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}=\dfrac{ v_{1}}{\Vert \mathbf{v}\Vert } \\ \cos \beta &=&\dfrac{v_{2}}{\sqrt{ v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}=\dfrac{v_{2}}{\Vert \mathbf{v}\Vert } \\ \cos \gamma &=&\dfrac{v_{3}}{\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}}=\dfrac{v_{3}}{\Vert \mathbf{v}\Vert } \end{eqnarray*}\]
• For a vector \(\mathbf{v}\) in space: \(\mathbf{v}=\left\Vert \mathbf{v} \right\Vert \left[ \cos \alpha \mathbf{i}+\cos \beta \mathbf{j}+\cos \gamma \mathbf{k}\right] \) (p. 719)
• Decomposition of a nonzero vector \(\mathbf{v}\): (p. 720) \[ \begin{eqnarray*} &&\hbox{Projection of } \mathbf{v} \hbox{ along } \mathbf{w} \hbox{ is } \mathbf{v}_{1}= \hbox{proj}_{\mathbf{w}}\mathbf{v}=\dfrac{\mathbf{v}\,{\cdot}\, \mathbf{ w}}{\left\Vert \mathbf{w}\right\Vert ^{2}}\mathbf{w}\\ &&\hbox{The vector \(\mathbf{v}_{2}\)} \hbox{ perpendicular to } \mathbf{w} \hbox{ is } \mathbf{v}_{2}=\mathbf{v}-\mathbf{v}_{1} \end{eqnarray*}\]

• Cross product: (p. 725)
  If \(\mathbf{v}\) \(=v_{1}\mathbf{i}+v_{2}\,\mathbf{j} +v_{3}\mathbf{k}\) and \(\mathbf{w}\) \(=w_{1}\mathbf{i}+w_{2} \mathbf{j}+w_{3} \mathbf{k}\), then \(\qquad \qquad \qquad \mathbf{v}\times \mathbf{w}=( v_{2}w_{3}-v_{3}w_{2}) \mathbf{i}-( v_{1}w_{3}-v_{3}w_{1}) \mathbf{j}+( v_{1}w_{2}-v_{2}w_{1}) \mathbf{k}\).
• Algebraic properties of the cross product: (p. 726) If \(\mathbf{u}\) ,\(\mathbf{v}\), and \(\mathbf{w}\) are vectors in space and if \(a\) is a scalar, then

• \(\mathbf{v}\times \mathbf{v}=\mathbf{0}\)
• \(\mathbf{v}\times \mathbf{w}=-(\mathbf{w}\times \mathbf{v})\)
• \(a\,(\mathbf{v}\times \mathbf{w})=(a\,\mathbf{v})\times \mathbf{w}=\mathbf{v}\times (a\mathbf{w})\)
• \(\mathbf{v}\times (\mathbf{w}+\mathbf{u})=(\mathbf{v}\times \mathbf{w})+(\mathbf{v}\times \mathbf{u})\)
• \(\left\Vert \mathbf{v}\times \mathbf{w}\right\Vert ^{2}=\left\Vert \mathbf{v}\right\Vert ^{2}\,\left\Vert \mathbf{w} \right\Vert ^{2}-(\mathbf{v}\,{\cdot}\, \mathbf{w})^{2}\)

• Geometric properties of the cross product: (p. 727)
  If \(\mathbf{v}\) and \(\mathbf{w}\) are nonzero vectors in space, then

• \(\mathbf{v}\times \mathbf{w}\) is orthogonal to both \(\mathbf{v}\) and \(\mathbf{w}\).
• \(\Vert \mathbf{v}\times \mathbf{w}\Vert =\Vert \mathbf{v}\Vert \Vert \mathbf{w}\Vert \sin \theta \), where \(\theta \) is the angle between \(\mathbf{v}\neq \mathbf{0}\) and \(\mathbf{w}\neq \mathbf{0}\).
• \(\Vert \mathbf{v}\times \mathbf{w}\Vert \) is the area of the parallelogram having \(\mathbf{v}\neq \mathbf{0}\) and \( \mathbf{w}\neq \mathbf{0}\) as adjacent sides.
• \(\mathbf{v}\times \mathbf{w}=\mathbf{0}\) if and only if \(\mathbf{v}\) and \(\mathbf{w}\) are parallel vectors.

Lines:

• Vector equation of a line: \(\mathbf{r}=\mathbf{r}_{0}+t\mathbf{D}\) (p. 733)
• Parametric equations of a line: \(x=x_{0}+at\), \(y=y_{0}+bt\), \( z=z_{0}+ct\) (p. 734)
• Symmetric equations of a line: \(\dfrac{x-x_{0}}{a}=\dfrac{y-y_{0}}{b} =\dfrac{z-z_{0}}{c}\)  where \({abc} \ne 0.\) (p. 735)

Planes:

• Vector equation of a plane: \(( \mathbf{r}-\mathbf{r}_{0}) \,{\cdot}\, \mathbf{N}=0,\) where \(\mathbf{N}\) is normal to the plane. (p. 737)
• General equation of a plane: \(Ax+By+Cz=D,\) where at least one number \(A, B, C\) is not \(0.\) (p. 738)
• Distance \(d\) from a point \(( x_{0},y_{0},z_{0}) \) to a plane \(Ax+By+Cz=D\): \(d=\dfrac{\left\vert Ax_{0}+By_{0}+Cz_{0}-D\right\vert }{\sqrt{A^{2}+B^{2}+C^{2}}}\) (p. 740)

• Ellipsoid: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) (p. 744)
• Elliptic cone: \(z^{2}=\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}\) (p. 745)
• Elliptic paraboloid: \(z=\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}\) (p. 745)
• Hyperbolic paraboloid: \(z=\dfrac{y^{2}}{b^{2}}-\dfrac{x^{2}}{a^{2}}\) (p. 746)
• Hyperboloid of one sheet: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}=1\) (p. 747)
• Hyperboloid of two sheets: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}=-1\) (p. 747)
• Parabolic cylinder: \(x^{2}=4ay\) (p. 747)
• Elliptic cylinder: \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \) (p. 749)
• Hyperbolic cylinder: \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) (p. 749)