Essential Concepts

• The dot product, or scalar product, of two vectors [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex] is [latex]{\bf{u}}\cdot{\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[/latex].
• The dot product satisfies the following properties:
  • [latex]{\bf{u}}\cdot{\bf{v}}={\bf{v}}\cdot{\bf{u}}[/latex]
  • [latex]{\bf{u}}\cdot\left({\bf{v}}+{\bf{w}}\right)={\bf{u}}\cdot{\bf{v}}+{\bf{u}}\cdot{\bf{w}}[/latex]
  • [latex]c({\bf{u}}\cdot{\bf{v}})=(c{\bf{u}})\cdot{\bf{v}}={\bf{u}}\cdot(c{\bf{v}})[/latex]
  • [latex]{\bf{v}}\cdot{\bf{v}}=\parallel{\bf{v}}\parallel^{2}[/latex]
• The dot product of two vectors can be expressed, alternatively, as [latex]{\bf{u}}\cdot{\bf{v}}=\parallel{\bf{u}}\parallel\parallel{\bf{v}}\parallel\cos\theta[/latex]. This form of the dot product is useful for finding the measure of the angle formed by two vectors.
• Vectors [latex]\bf{u}[/latex] and [latex]\bf{v}[/latex] are orthogonal if [latex]{\bf{u}}\cdot{\bf{v}}=0[/latex].
• 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 [latex]\bf{v}[/latex] onto [latex]\bf{u}[/latex] is the vector [latex]\text{proj}_{\bf{u}}{\bf{v}}=\frac{\bf{u}\cdot\bf{v}}{\parallel{\bf{u}}\parallel^{2}}{\bf{u}}[/latex]. The magnitude of this vector is known as the scalar projection of [latex]\bf{v}[/latex] onto [latex]\bf{u}[/latex], given by [latex]\text{comp}_{\bf{u}}{\bf{v}}=\frac{\bf{u}\cdot\bf{v}}{\parallel{\bf{u}}\parallel}[/latex].
• Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector [latex]\bf{F}[/latex] and the displacement is represented by the vector [latex]\bf{s}[/latex], then the work done [latex]W[/latex] is given by the formula [latex]W={\bf{F}}\cdot{\bf{s}}=\parallel{\bf{F}}\parallel\parallel{\bf{s}}\parallel\cos\theta[/latex].