Find the vector v that has a magnitude of 9 and is in the same direction as u, where $\mathbf { u } = \langle 6 , - 5 \rangle$ .
A) $\mathbf { v } = \left\langle \frac { 5 \sqrt { 61 } } { 61 } , - \frac { 6 \sqrt { 61 } } { 61 } \right\rangle$
B) $\mathrm { v } = \left\langle \frac { 45 \sqrt { 61 } } { 61 } , - \frac { 54 \sqrt { 61 } } { 61 } \right\rangle$
C) $\mathbf { v } = \left\langle \frac { \sqrt { 61 } } { 61 } , - \frac { \sqrt { 61 } } { 61 } \right\rangle$
D) $\mathbf { v } = \left\langle \frac { 6 \sqrt { 61 } } { 61 } , - \frac { 5 \sqrt { 61 } } { 61 } \right\rangle$
E) $\mathbf { v } = \left\langle \frac { 54 \sqrt { 61 } } { 61 } , - \frac { 45 \sqrt { 61 } } { 61 } \right\rangle$

Question

Find the vector v that has a magnitude of 9 and is in the same direction as u, where $\mathbf { u } = \langle 6 , - 5 \rangle$ .  
A)  $\mathbf { v } = \left\langle \frac { 5 \sqrt { 61 } } { 61 } , - \frac { 6 \sqrt { 61 } } { 61 } \right\rangle$
B)  $\mathrm { v } = \left\langle \frac { 45 \sqrt { 61 } } { 61 } , - \frac { 54 \sqrt { 61 } } { 61 } \right\rangle$
C)  $\mathbf { v } = \left\langle \frac { \sqrt { 61 } } { 61 } , - \frac { \sqrt { 61 } } { 61 } \right\rangle$
D)  $\mathbf { v } = \left\langle \frac { 6 \sqrt { 61 } } { 61 } , - \frac { 5 \sqrt { 61 } } { 61 } \right\rangle$
E)  $\mathbf { v } = \left\langle \frac { 54 \sqrt { 61 } } { 61 } , - \frac { 45 \sqrt { 61 } } { 61 } \right\rangle$

Quizplus · Accepted Answer

To find a vector with a specific magnitude in the same direction as another vector, you first normalize the original vector and then multiply it by the desired magnitude. The magnitude of $\mathbf{u} = \langle 6, -5 \rangle$ is $\sqrt{6^2 + (-5)^2} = \sqrt{61}$. The normalized vector of $\mathbf{u}$ is $\left\langle \frac{6}{\sqrt{61}}, \frac{-5}{\sqrt{61}} \right\rangle$. Multiplying this by 9 (the desired magnitude) gives $\mathbf{v} = \left\langle \frac{54\sqrt{61}}{61}, -\frac{45\sqrt{61}}{61} \right\rangle$, which matches option E.

Find the Vector V That Has a Magnitude of 9 u=⟨6,−5⟩\mathbf { u } = \langle 6 , - 5 \rangleu=⟨6,−5⟩

Find the Vector V That Has a Magnitude of 9 $\mathbf { u } = \langle 6 , - 5 \rangle$