the vectors u and v have the same direction.\na. find ||u||.\nb. find ||v||.\nc. is u = v? explain.\na…

Explanation:

Step1: Recall vector - magnitude formula

The magnitude of a vector $\mathbf{u}=(x_2 - x_1,y_2 - y_1)$ with initial point $(x_1,y_1)$ and terminal point $(x_2,y_2)$ is given by $|\mathbf{u}|=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For vector $\mathbf{u}$ with initial point $(- 2,1)$ and terminal point $(5,3)$, we have $x_1=-2,y_1 = 1,x_2 = 5,y_2=3$.

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

Substitute the values into the formula: [ \begin{align*} |\mathbf{u}|&=\sqrt{(5-(-2))^2+(3 - 1)^2}\ &=\sqrt{(5 + 2)^2+(3 - 1)^2}\ &=\sqrt{7^2+2^2}\ &=\sqrt{49 + 4}\ &=\sqrt{53} \end{align*} ]

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

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

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

Two vectors $\mathbf{u}$ and $\mathbf{v}$ are equal if they have the same magnitude and direction. Since $\mathbf{u}$ and $\mathbf{v}$ have the same direction (given) and $|\mathbf{u}|=\sqrt{53}$ and $|\mathbf{v}|=\sqrt{53}$, they have the same magnitude. Also, if we consider the component - form of vectors, $\mathbf{u}=(5-(-2),3 - 1)=(7,2)$ and $\mathbf{v}=(7 - 0,2 - 0)=(7,2)$. So $\mathbf{u}=\mathbf{v}$.

Answer:

a. $\sqrt{53}$ b. $\sqrt{53}$ c. Yes. They have the same magnitude ($\sqrt{53}$) and direction, and their component - forms are equal ($\mathbf{u}=(7,2)$ and $\mathbf{v}=(7,2)$).