Question

To find a dot product \(u \cdot v\) and \(u \cdot w\).

Short answer

The dot product of\({\rm{u}} \cdot {\rm{v}}\) is v and \({\rm{u}} \cdot {\rm{w}}\) is \( - \frac{1}{2}\).

Step-by-step solution

Concept of Dot Product

Formula Used:

Write the expression to find\({\rm{u}} \cdot {\rm{v}}\)in terms of\(\theta \).

\({\rm{u}} \cdot {\rm{v}} = |{\rm{u}}||{\rm{v}}|\cos \theta \)

Here,

\(|u|\)is the magnitude of vector\(u\),

\(|{\rm{v}}|\)is the magnitude of vector\(v\), and

\(\theta \)is the angle between vectors\(u\)and\(v\).

Calculation of the dot product of\({\rm{u}} \cdot {\rm{v}}\)

Consider \({\bf{u}},{\bf{v}}\) and \({\bf{w}}\) are the unit vectors. In an equilateral triangle, the angle between any two sides is \({60^\circ }\).

As \({\bf{u}},{\bf{v}}\) and \({\bf{w}}\)are the unit vectors, the values of \(\left| u \right|,\left| v \right|\)and \(|{\rm{w}}|\) is \(1\) .

From the equilateral triangle, the angle between \(u\) and \(v\) is \({60^\circ }\).

In equation\({\rm{u}} \cdot {\rm{v}}\), substitute \(1\) for \(|{\rm{u}}|,1\) for \(|{\rm{v}}|\) and \({60^\circ }\) for \(\theta \).

\(\begin{aligned}{l}{\rm{u}} \cdot {\rm{v}} &= (1)(1)\cos \left( {{{60}^\circ }} \right)\\{\rm{u}} \cdot {\rm{v}} &= 1\left( {\frac{1}{2}} \right)\\{\rm{u}} \cdot {\rm{v}} &= \frac{1}{2}\end{aligned}\)

Thus, \({\rm{u}} \cdot {\rm{v}}\) is \(\frac{{\rm{I}}}{2}\).

Calculation of the dot product\({\rm{u}} \cdot w\)

Rewrite equation of\({\rm{u}} \cdot {\rm{v}}\),

\({\rm{u}} \cdot {\rm{w}} = |{\rm{u}}||{\rm{w}}|\cos \theta \)

From the equilateral triangle, the angle between \({\rm{u}}\) and \({\rm{w}}\) is \({120^\circ }\).

In equation\({\rm{u}} \cdot {\rm{w}}\) , substitute \(1\) for \(|{\rm{u}}|,1\) for \(|{\rm{w}}|\) and \({120^\circ }\) for \(\theta \).

\(\begin{aligned}{l}{\rm{u}} \cdot {\rm{w}} &= (1)(1)\cos \left( {{{120}^\circ }} \right)\\{\rm{u}} \cdot {\rm{w}} &= {\rm{ (1) }}\left( { - \frac{1}{2}} \right)\\{\rm{u}} \cdot {\rm{w}} &= - \frac{1}{2}\end{aligned}\)

Thus, \({\rm{u}} \cdot {\rm{w}}\) is \( - \frac{1}{2}\).