Essential Concepts

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