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.
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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
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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???.
