Find the vector $\mathbf { v }$ that has a magnitude of 4 and is in the same direction as $\mathbf { u }$, where $\mathbf { u } = \langle - 5,4 \rangle$.
A) $\mathbf { v } = \left\langle \frac { - 20 } { \sqrt { 41 } } , \frac { 16 } { \sqrt { 41 } } \right\rangle$
B) $\mathbf { v } = \left\langle \frac { - 5 } { \sqrt { 41 } } , \frac { 4 } { \sqrt { 41 } } \right\rangle$
C) $\mathbf { v } = \left\langle \frac { - 4 } { \sqrt { 41 } } , \frac { 1 } { \sqrt { 41 } } \right\rangle$
D) $\mathbf { v } = \left\langle \frac { - 1 } { \sqrt { 41 } } , \frac { 1 } { \sqrt { 41 } } \right\rangle$
E) $ \mathbf { v } = \left\langle \frac { - 4 } { \sqrt { 41 } } , \frac { 5 } { \sqrt { 41 } } \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 $\mathbf{u} = \langle -5, 4 \rangle$ is $\sqrt{(-5)^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41}$. The normalized vector of $\mathbf{u}$ is $\left\langle \frac{-5}{\sqrt{41}}, \frac{4}{\sqrt{41}} \right\rangle$. Multiplying this by 4 gives $\mathbf{v} = \left\langle \frac{-20}{\sqrt{41}}, \frac{16}{\sqrt{41}} \right\rangle$, which matches choice A.

Find the Vector v\mathbf { v }v That Has a Magnitude of 4 and Is in the Magnitude

Find the Vector $\mathbf { v }$ That Has a Magnitude of 4 and Is in the Magnitude