Find the vector $\mathbf { v }$ that has a magnitude of 6 and is in the same direction as $\mathbf { u }$, where $\mathbf { u } = \langle - 6 , - 5 \rangle$.
A) $\mathbf { v } = \left\langle \frac { - 36 } { \sqrt { 61 } } , \frac { - 30 } { \sqrt { 61 } } \right\rangle$
B) $\mathbf { v } = \left\langle \frac { - 6 } { \sqrt { 61 } } , \frac { - 5 } { \sqrt { 61 } } \right\rangle$
C) $\mathbf { v } = \left\langle \frac { - 1 } { \sqrt { 61 } } , \frac { - 6 } { \sqrt { 61 } } \right\rangle$
D) $\mathbf { v } = \left\langle \frac { - 1 } { \sqrt { 61 } } , \frac { 1 } { \sqrt { 61 } } \right\rangle$
E) $\quad \mathbf { v } = \left\langle \frac { 5 } { \sqrt { 61 } } , \frac { 6 } { \sqrt { 61 } } \right\rangle$

Question

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 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 6 (the desired magnitude) gives $\mathbf{v} = \left\langle \frac{-36}{\sqrt{61}}, \frac{-30}{\sqrt{61}} \right\rangle$, which matches choice A.

Find the Vector v\mathbf { v }v That Has a Magnitude of 6 and Is in the Magnitude

Find the Vector $\mathbf { v }$ That Has a Magnitude of 6 and Is in the Magnitude