Dot Product And Its Applications

Dot Product

The dot product of two vectors results in a scalar. The dot product can be calculated in either of the following ways.

(1)
\begin{align} \vec{u} \cdot \vec{v}=u_xv_x+u_yv_y+u_zv_z \end{align}
(2)
\begin{align} \vec{u} \cdot \vec{v}=|\vec{u}||\vec{v}|cos \theta \end{align}

Where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$

Exercise
Find the dot product of $\vec{u}=\left < 1,4,-2 \right >$ and $\vec{v}=\left <-7,0,8 \right >$

Exercise
Find the dot product of $\vec{u}=20@80^\circ$ and $\vec{v}=12@10^\circ$

The Angle Between Two Vectors

The angle $0^\circ \leq \theta \leq 180^\circ$ made by the vectors $\vec{u}$ and $\vec{v}$ when they are placed with their initial points coinciding is given by

(3)
\begin{align} \cos \theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}||\vec{v}|} \end{align}

Therefore,

(4)
\begin{align} \theta = \cos^{-1} \left ( \frac{\vec{u} \cdot \vec{v}}{|\vec{u}||\vec{v}|} \right ) \end{align}

Recall that arccosine has a range from $0^\circ$ to $180^\circ$, which is exactly the range of values that the angle between two vectors can take on. Therefore, there is no need for any quadrant checks with this calculation.

Exercise
Find the angle between $\vec{u}=\left < -2,6,10 \right >$ and $\vec{v}=\left <2,4,0 \right >$

Perpendicular Vectors

Two vectors are perpendicular if their dot product equals zero. This can easily be seen from the above Eq.(2). When two vectors are perpendicular, the anlge between them is $90^\circ$. We know that $\cos 90^\circ=0$. Therefore, the dot product will be,

(5)
\begin{align} \vec{u} \cdot \vec{v} = (|\vec{u}||\vec{v}|)(\cos 90^\circ)=|\vec{u}||\vec{v}|(0)=0 \end{align}

We can take this analysis a step further. Also recall that $-1 \leq \cos \theta \leq 1$. Therefore, the dot product will be largest and equal to $|\vec{u}||\vec{v}|$ when $\cos \theta=1$ or when the angle between the two vectors is $0^\circ$. In other words, when the vectors are pointing in the same direction.Likewise, the dot product will be smallest and equal to $-|\vec{u}||\vec{v}|$ when $\cos \theta=-1$ or when the angle between the two vectors is $180^\circ$. In other words, when the vectors are pointing in opposite directions.

Exercise
Show that $\vec{u}=\left < -3,5,-1 \right >$ and $\vec{v}=\left <1,2,7 \right >$ are perpendicular

Exercise
Determine the value of $a$ such that $\vec{u}=\left < 4,5,6 \right >$ and $\vec{v}=\left <2,a,-3 \right >$ are perpendicular

Exercises

  1. Find the dot product of the vectors $<6, 8, –5>$ and $<–7, 3, 4>$.
  2. Find the dot product of the vectors 27 @ 35º and 19 @ 46º.
  3. Find the angle between the vectors $4\hat{x}+9\hat{y}$ and $-3\hat{x}+5\hat{y}$.
  4. Find the angle between the vectors $<6, 8, –5>$ and $<–7, 3, 4>$.
  5. Are the vectors $4\hat{x}+9\hat{y}$ and $-7\hat{x}+3\hat{y}$ perpendicular? Why or why not?
  6. Are the vectors $<8,6,-2>$ and $<7,5,13>$ perpendicular? Why or why not?
  7. Are the vectors $<8,6,-2>$ and $<-4,5,-1>$ perpendicular? Why or why not?
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License