Question

NO LINKS OR ANSWERING QUESTIONS YOU DON'T KNOW!!! THIS IS NOT A TEST OR AN ASSESSMENT!! Please help me with these math questions. Chapter 14 part 2

3. What are two ways that a vector can be represented?

4. While playing basketball, Sofia takes notes of the extreme vectors for which she bats. What is the angle between her two extreme bats? Extreme: <-8, 12>
Extreme 2: <13, 15> SHOW YOUR WORK!!!!

Answer 1

See below

Explanation:

3. What are two ways that a vector can be represented?

Considering a vector
`\vec{v}` in some vector space
`\mathbb R^n` we have

`\vec{v} = \langle a,b\rangle`

This is the component form. I don't like that way. It is probably used in high school, but

`\vec{v} = \begin{pmatrix} a\\ b\\ \end{pmatrix}`

is preferable because the inner product on
`\mathbb R^n` is defined to be

`$\langle a,b\rangle := \sum_(i = 1)^n a_i b_i$`

You can also write it using linear form such as
`\vec{v} = 2i+2j`

4.

For this question, I think you meant

vectors

`\vec{u_1} = (-8, 12)`

`\vec{u_2} = (13, 15)`

Once

`\cos(\theta)=\frac{\vec{u_1} \cdot\vec{u_2}}{||\vec{u_1}||||\vec{u_2}||}`

Considering that the dot product is

`\vec{u_1}\cdot \vec{u_2} = (-8)\cdot 13 + 12\cdot 15 = -104+180= 76`

and the norm of
`\vec{u_1}` is
`||\vec{u_1}|| = √((-8)^2 + 12^2) = √(64 + 144)= √(208)`

and the norm of
`\vec{u_2}` is
`||\vec{u_2}|| = √(13^2 + 15^2) = √(169 + 225)= √(394)`

Thus,

`\cos(\theta)=(76)/(√(208) √(394)) = (19)/(√(13)√(394))=(19)/(√(5122))`

`\therefore \theta = \arccos \left((19)/(√(5122)) \right)`