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 \(0.5\) and \({\rm{u}} \cdot {\rm{w}}\) is \(0\).

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 the right angle triangle formed by \({\bf{u}}\) and \(v\).

Write the expression to find magnitude of \(v(|v|)\).

\(|v| = |u|\cos \theta \)

As \({\bf{u}}\), and \({\bf{w}}\) are the unit vectors, the value of \(|u|\)and \(|{\bf{w}}|\) is\(1\).

The angle between \({\bf{u}}\) and \(v\) is \({45^\circ }\).

In equation of\(|v|\), substitute\(1\) for \(|{\rm{u}}|\) and \({45^\circ }\) for \(\theta \).

\(\begin{aligned}{l}|v| &= (1)\cos \left( {{{45}^\circ }} \right)\\|v| &= (1)(0.707)\\|v| &= 0.707\end{aligned}\)

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

In equation of\({\rm{u}} \cdot {\rm{v}}\), substitute\(1\)for \(|{\rm{u}}|,0.707\) for \(|{\rm{v}}|\) and \({45^\circ }\) for \(\theta \).

\(\begin{aligned}{l}{\rm{u}} \cdot {\rm{v}} &= (1)(0.707)\cos \left( {{{45}^\circ }} \right)\\{\rm{u}} \cdot {\rm{v}} &= (0.707)(0.707)\\{\rm{u}} \cdot {\rm{v}} &= 0.49\\{\rm{u}} \cdot {\rm{v}} \cong 0.5\end{aligned}\)

Thus, \({\rm{u}} \cdot {\rm{v}}\) is \(0.5.\)

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

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

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

Substitute \(1\) for \(|{\rm{u}}|,1\) for \(|{\rm{w}}|\) and \({90^\circ }\) for \(\theta \).

\(\begin{aligned}{l}{\rm{u}} \cdot {\rm{w}} &= (1)(1)\cos \left( {{{90}^\circ }} \right)\\{\rm{u}} \cdot {\rm{w}} &= (1)(0)\\{\rm{u}} \cdot {\rm{w}} &= 0\end{aligned}\)

Thus, \({\rm{u}} \cdot {\rm{w}}\) is\(0\) .