Assume that the vectors $\mathbf { a }$ , $\mathbf { b }$ , $\text { C }$ and $\text { d }$ are defined as follows: $\mathbf { a } = \langle 1,2 \rangle$ $$\mathbf { b } = \langle 5,4 \rangle$$ $\mathbf { c } = \langle 5 , - 2 \rangle$
$\mathbf { d } = \langle - 2,0 \rangle$ Compute $\frac { 1 } { | 3 b - 4 d | } ( 3 b - 4 c )$
A) $\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$
B) $\left\langle \frac { 20 \sqrt { 673 } } { 673 } , \frac { 5 \sqrt { 673 } } { 673 } \right\rangle$
C) $\left\langle - \frac { 20 \sqrt { 673 } } { 673 } , - \frac { 5 \sqrt { 673 } } { 673 } \right\rangle$
D) $\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , - \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$
E) $\left\langle \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$

Question

Assume that the vectors $\mathbf { a }$ , $\mathbf { b }$ , $\text { C }$ and $\text { d }$ are defined as follows: $\mathbf { a } = \langle 1,2 \rangle$ $$\mathbf { b } = \langle 5,4 \rangle$$ $\mathbf { c } = \langle 5 , - 2 \rangle$ 
$\mathbf { d } = \langle - 2,0 \rangle$ Compute $\frac { 1 } { | 3 b - 4 d | } ( 3 b - 4 c )$
A) $\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$
B) $\left\langle \frac { 20 \sqrt { 673 } } { 673 } , \frac { 5 \sqrt { 673 } } { 673 } \right\rangle$
C) $\left\langle - \frac { 20 \sqrt { 673 } } { 673 } , - \frac { 5 \sqrt { 673 } } { 673 } \right\rangle$
D) $\left\langle - \frac { 5 \sqrt { 673 } } { 673 } , - \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$
E) $\left\langle \frac { 5 \sqrt { 673 } } { 673 } , \frac { 20 \sqrt { 673 } } { 673 } \right\rangle$

Quizplus · Accepted Answer

First, calculate $3\mathbf{b} - 4\mathbf{d}$ and $3\mathbf{b} - 4\mathbf{c}$. For $3\mathbf{b} - 4\mathbf{d}$, we have $3\langle 5,4 \rangle - 4\langle -2,0 \rangle = \langle 15,12 \rangle + \langle 8,0 \rangle = \langle 23,12 \rangle$. The magnitude $|\langle 23,12 \rangle|$ is $\sqrt{23^2 + 12^2} = \sqrt{529 + 144} = \sqrt{673}$. For $3\mathbf{b} - 4\mathbf{c}$, we have $3\langle 5,4 \rangle - 4\langle 5,-2 \rangle = \langle 15,12 \rangle - \langle 20,-8 \rangle = \langle -5,20 \rangle$. Thus, $\frac{1}{|\langle 23,12 \rangle|}\langle -5,20 \rangle = \frac{1}{\sqrt{673}}\langle -5,20 \rangle = \left\langle -\frac{5\sqrt{673}}{673}, \frac{20\sqrt{673}}{673} \right\rangle$.

Assume That the Vectors a\mathbf { a }a b\mathbf { b }b

Assume That the Vectors $\mathbf { a }$ $\mathbf { b }$