Question

To find a dot product between \({\rm{a}}\) and \({\rm{b}}\).

Short answer

The dot product between two vectors \({\rm{a}}\) and \({\rm{b}}\) is \({\rm{a}} \cdot {\rm{b}} = 3\sqrt 3 \).

Step-by-step solution

Concept of Dot Product

Formula used:

The dot product of two vectors\({\rm{a}}\)and\({\rm{b}}\)in terms of\(\theta \)is\({\rm{a}} \cdot {\rm{b}} = |{\rm{a}}||{\rm{b}}|\cos \theta \).

Calculation of the dot product

The given magnitude of two vectors are \(|{\rm{a}}| = 3\) and \(|{\rm{b}}| = \sqrt 6 \) with angle \({45^\circ }\) between them.

Substitute \(|{\rm{a}}| = 3,|\;{\rm{b}}| = \sqrt 6 \), and \(\theta = {45^\circ }\) in the above formula and obtain the dot product as follows.

\(\begin{aligned}{l}a \cdot b &= (3)(\sqrt 6 )\cos \left( {{{45}^\circ }} \right)\\a \cdot b &= 3\sqrt 6 \left( {\frac{1}{{\sqrt 2 }}} \right)\\a \cdot b &= 3\sqrt 3 \left( {\frac{{\sqrt 2 }}{{\sqrt 2 }}} \right)\\a \cdot b &= 3\sqrt 3 \end{aligned}\)

Thus, the dot product between two vector \(a\) and \(b\) is \({\rm{a}} \cdot {\rm{b}} = 3\sqrt 3 \).