Question

Someone please give me the answer for this

Answer 1

The magnitude of
`\overrightarrow{AC}` is
`√(85)`.

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}\| = √(85)`