the vectors u and v have the same direction. a. find ||u||. b. find ||v||. c. is u = v? explain. a. ||u|| =…

Explanation:

Step1: Recall vector - magnitude formula

The magnitude of a vector $\vec{u}=(x_2 - x_1,y_2 - y_1)$ is given by $|\vec{u}|=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For vector $\vec{u}$ with initial point $(-1,2)$ and terminal point $(6,4)$, we have $x_1=-1,y_1 = 2,x_2=6,y_2 = 4$.

Step2: Calculate $|\vec{u}|$

[ \begin{align*} |\vec{u}|&=\sqrt{(6-(-1))^2+(4 - 2)^2}\ &=\sqrt{(6 + 1)^2+2^2}\ &=\sqrt{7^2+2^2}\ &=\sqrt{49+4}\ &=\sqrt{53} \end{align*} ]

Step3: Calculate $|\vec{v}|$

For vector $\vec{v}$ with initial point $(0,0)$ and terminal point $(7,2)$, using the magnitude formula $|\vec{v}|=\sqrt{(7 - 0)^2+(2-0)^2}=\sqrt{7^2+2^2}=\sqrt{49 + 4}=\sqrt{53}$.

Step4: Determine if $\vec{u}=\vec{v}$

Two vectors $\vec{u}=(x_1,y_1)$ and $\vec{v}=(x_2,y_2)$ are equal if they have the same magnitude and direction. Since $\vec{u}$ has initial - point $(-1,2)$ and terminal - point $(6,4)$ and $\vec{v}$ has initial - point $(0,0)$ and terminal - point $(7,2)$, although $|\vec{u}|=|\vec{v}|=\sqrt{53}$ and they have the same direction, they have different initial points. So $\vec{u}\neq\vec{v}$.

Answer:

a. $\sqrt{53}$ b. $\sqrt{53}$ c. No. Although they have the same magnitude and direction, they have different initial points.