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Chapter 6: Q10E (page 331)

In Exercises 9-12, find a unit vector in the direction of the given vector.

10. \(\left( {\begin{aligned}{*{20}{c}}{ - 6}\\4\\{ - 3}\end{aligned}} \right)\)

Short Answer

Expert verified

The unit vector \({\mathop{\rm u}\nolimits} \) in the direction of v is \({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{\frac{{ - 6}}{{\sqrt {61} }}}\\{\frac{4}{{\sqrt {61} }}}\\{\frac{{ - 3}}{{\sqrt {61} }}}\end{aligned}} \right)\).

Step by step solution

01

Definition of a unit vector

Aunit vectoris a vector with a length of 1. When dividing a nonzero vector v by its length, namely, multiply by \(\frac{1}{{\left\| {\mathop{\rm v}\nolimits} \right\|}}\)-, we get a unit vector u since \(\left( {\frac{1}{{\left\| {\mathop{\rm v}\nolimits} \right\|}}} \right)\left\| {\mathop{\rm v}\nolimits} \right\|\) is the length of u. The process of producing u from v is known as thenormalizingv, and we describe that u is in the same direction as v.

02

Determine the unit vector in the direction

It is given that \({\mathop{\rm v}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 6}\\4\\{ - 3}\end{aligned}} \right)\).

Compute the length of \({\mathop{\rm v}\nolimits} \) as shown below:

\(\begin{aligned}{c}\left\| {\mathop{\rm v}\nolimits} \right\| &= \sqrt {{\mathop{\rm v}\nolimits} \cdot {\mathop{\rm v}\nolimits} } \\ &= \sqrt {{{\left( { - 6} \right)}^2} + {4^2} + {{\left( { - 3} \right)}^2}} \\ &= \sqrt {36 + 16 + 9} \\ &= \sqrt {61} \end{aligned}\)

Multiply v by \(\frac{1}{{\left\| {\mathop{\rm v}\nolimits} \right\|}}\) to obtain the unit vector \({\mathop{\rm u}\nolimits} \) as shown below:

\(\begin{aligned}{c}{\mathop{\rm u}\nolimits} &= \frac{1}{{\left\| {\mathop{\rm v}\nolimits} \right\|}}{\mathop{\rm v}\nolimits} \\ &= \frac{1}{{\sqrt {61} }}\left( {\begin{aligned}{*{20}{c}}{ - 6}\\4\\{ - 3}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{*{20}{c}}{\frac{{ - 6}}{{\sqrt {61} }}}\\{\frac{4}{{\sqrt {61} }}}\\{\frac{{ - 3}}{{\sqrt {61} }}}\end{aligned}} \right)\end{aligned}\)

Thus, the unit vector \({\mathop{\rm u}\nolimits} \) in the direction of v is \({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{\frac{{ - 6}}{{\sqrt {61} }}}\\{\frac{4}{{\sqrt {61} }}}\\{\frac{{ - 3}}{{\sqrt {61} }}}\end{aligned}} \right)\).

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