2.3 The dot product  (Page 7/16)

We have

In U.S. standard units, we measure the magnitude of force $\Vert \text{F}\Vert$ in pounds. The magnitude of the displacement vector $\Vert \overrightarrow{PQ}\Vert$ tells us how far the object moved, and it is measured in feet. The customary unit of measure for work, then, is the foot-pound. One foot-pound is the amount of work required to move an object weighing 1 lb a distance of 1 ft straight up. In the metric system, the unit of measure for force is the newton (N), and the unit of measure of magnitude for work is a newton-meter (N·m), or a joule (J).

A constant force of 30 lb is applied at an angle of 60° to pull a handcart 10 ft across the ground ( [link] ). What is the work done by this force?

150 ft-lb

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Key concepts

• The dot product, or scalar product, of two vectors $\text{u}=⟨u_1,u_2,u_3⟩$ and $\text{v}=⟨v_1,v_2,v_3⟩$ is $\text{u}·\text{v}=u_1{v}_1+u_2{v}_2+u_3{v}_3.$
• The dot product satisfies the following properties:
  • $\text{u}·\text{v}=\text{v}·\text{u}$
  • $\text{u}·\left(\text{v}+\text{w}\right)=\text{u}·\text{v}+\text{u}·\text{w}$
  • $c\left(\text{u}·\text{v}\right)=\left(c\text{u}\right)·\text{v}=\text{u}·\left(c\text{v}\right)$
  • $\text{v}·\text{v}={\Vert \text{v}\Vert }^2$
• The dot product of two vectors can be expressed, alternatively, as $\text{u}·\text{v}=\Vert \text{u}\Vert \Vert \text{v}\Vert \text{cos}\theta .$ 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 $\text{u}·\text{v}=0.$
• 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 ${\text{proj}}_{\text{u}}\text{v}=\frac{\text{u}·\text{v}}{{\Vert \text{u}\Vert }^2}\text{u}.$ The magnitude of this vector is known as the scalar projection of v onto u , given by ${\text{comp}}_{\text{u}}\text{v}=\frac{\text{u}·\text{v}}{\Vert \text{u}\Vert }.$
• 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 $W=\text{F}·s=\Vert \text{F}\Vert \Vert \text{s}\Vert \text{cos}\theta .$

Key equations

• Dot product of u and v
  $\begin{array}{cc}\hfill \text{u}·\text{v}& =u_1{v}_1+u_2{v}_2+u_3{v}_3\hfill \\ & =\Vert \text{u}\Vert \Vert \text{v}\Vert \text{cos}\theta \hfill \end{array}$
• Cosine of the angle formed by u and v
  $\text{cos}\theta =\frac{\text{u}·\text{v}}{\Vert \text{u}\Vert \Vert \text{v}\Vert }$
• Vector projection of v onto u
  ${\text{proj}}_{\text{u}}\text{v}=\frac{\text{u}·\text{v}}{{\Vert \text{u}\Vert }^2}\text{u}$
• Scalar projection of v onto u
  ${\text{comp}}_{\text{u}}\text{v}=\frac{\text{u}·\text{v}}{\Vert \text{u}\Vert }$
• Work done by a force F to move an object through displacement vector $\overrightarrow{PQ}$
  $W=\text{F}·\overrightarrow{PQ}=\Vert \text{F}\Vert \Vert \overrightarrow{PQ}\Vert \text{cos}\theta$

For the following exercises, the vectors u and v are given. Calculate the dot product $\text{u}·\text{v}.$

For the following exercises, the vectors a , b , and c are given. Determine the vectors $\left(\text{a}·\mathbf{\text{b}}\right)\mathbf{\text{c}}$ and $\left(\text{a}·\mathbf{\text{c}}\right)\mathbf{\text{b}}.$ Express the vectors in component form.