Dot product of two vectors

How to calculate a dot product

To take the dot product of two vectors $a$ and $b$, we multiply the vectors’ like coordinates and then add the products together. In other words, we multiply the $x$ coordinates of the two vectors, then add this to the product of the $y$ coordinates. If we have vectors in three-dimensional space, we’ll add the product of the $z$ coordinates as well.

Hi! I'm krista.

I create online courses to help you rock your math class. Read more.

If we’re given the vectors $a\langle a_1,a_2\rangle$ and $b\langle b_1,b_2\rangle$, then the dot product of $a$ and $b$ will be

$a\cdot{b}=a_1b_1+a_2b_2$

Finding the dot product of two vectors

---

Take the course

Want to learn more about Calculus 3? I have a step-by-step course for that. :)

---

The dot product in two dimensions

Example

Find the dot product the vectors.

$a=\langle2,1\rangle$

$b=\langle6,-2\rangle$

To find the dot product of the vectors $a$ and $b$, we’ll multiply like coordinates and then add the products together.

$a\cdot{b}=(2)(6)+(1)(-2)$

$a\cdot{b}=12-2$

$a\cdot{b}=10$

The dot product of the vectors $a$ and $b$ is $a\cdot{b}=10$.

To take the dot product of two vectors ???a??? and ???b???, we multiply the vectors’ like coordinates and then add the products together.

Example

Find the dot product the vectors.

$c=2i-6j+k$

$d=2j-3k$

Converting our vectors into standard form, we get

$c=\langle2,-6,1\rangle$

$d=\langle0,2,-3\rangle$

To find the dot product of the vectors $c$ and $d$, we’ll multiply like coordinates and then add the products together.

$c\cdot{d}=(2)(0)+(-6)(2)+(1)(-3)$

$c\cdot{d}=0-12-3$

$c\cdot{d}=-15$

The dot product of the vectors $c$ and $d$ is $c\cdot{d}=-15$.

---

Get access to the complete Calculus 3 course

---