Question

To determine the dot product between two vector \({\rm{a}}\) and \({\rm{b}}\).

Short answer

The dot product between two vector \({\rm{a}}\) and \({\rm{b}}\) is \({\rm{a}} \cdot {\rm{b}} = 14\).

Step-by-step solution

Concept of Dot Product

Consider a general expression to find the dot product between two two-dimensional vectors.

\(\begin{aligned}{l}{\rm{a}} \cdot {\rm{b}} &= \left\langle {{a_1},{a_2}} \right\rangle \cdot \left\langle {{b_1},{b_2}} \right\rangle \\{\rm{a}} \cdot {\rm{b}} &= {a_1}{b_1} + {a_2}{b_2}\end{aligned}\)

Calculation of the dot product between two vector

Note that, the minimum of two vectors are required to perform a dot product. The resultant dot product of two vectors is scalar. hence, the dot product is also known as a scalar product.

The given two vectors are \({\rm{a}} = \left\langle { - 2,\frac{1}{3}} \right\rangle \) and \({\rm{b}} = \langle - 5,12\rangle \).

Substitute the value of components \({a_1} = - 2,{a_2} = \frac{1}{3},{b_1} = - 5\), and \({b_2} = 12\) in the above result.

\(\begin{aligned}{l}a \cdot b &= ( - 2)( - 5) + \left( {\frac{1}{3}} \right)(12)\\a \cdot b &= 10 + 4\\a \cdot b &= 14\end{aligned}\)

Thus, the dot product between two vector \({\rm{a}}\) and \({\rm{b}}\) is \({\rm{a}} \cdot {\rm{b}} = 14\).