Question

someone please give me the answer for this

Answer 1

Answer:

The magnitude of $\overrightarrow{AC}$ is $\sqrt{85}$.

Step-by-step explanation:

There are two vectors: $\overrightarrow {AB} = \left(\begin{array}{ccc}6\\-9\end{array} \right)$, $\overrightarrow{CB} = \left(\begin{array}{ccc}1\\3\end{array}\right)$. From Linear Algebra, we have the following expressions:

$\overrightarrow{AC} = \vec C - \vec A$

$\overrightarrow{AC} = (\vec C - \vec B) + (\vec B - \vec A)$

$\overrightarrow{CB} = -\overrightarrow {BC}$

$\overrightarrow{BC} = - \overrightarrow{CB}$

$\vec C - \vec B = -\overrightarrow {CB}$

$\overrightarrow{AB} = \vec B - \vec A$

Then,

$\overrightarrow{AC} = -\overrightarrow{CB}+\overrightarrow{AB}$

$\overrightarrow{AC} = -\left(\begin{array}{ccc}1\\3\end{array}\right)+\left(\begin{array}{ccc}6\\-9\end{array}\right)$

$\overrightarrow{AC} = \left(\begin{array}{ccc}7\\-6\end{array}\right)$

The magnitude of $\overrightarrow{AC}$ is:

$\|\overrightarrow{AC}\| = \sqrt{\overrightarrow{AC}\,\bullet\,\overrightarrow{AC}}$

$\|\overrightarrow{AC}\| = \sqrt{7^{2}+(-6)^{2}}$

$\|\overrightarrow{AC}\| = \sqrt{85}$