Question

Correct answers only and answer all of them please
Put number by answer ex:(1:A, 2:B, 3:C,.....etc)
I'll mark brainliest!!

Answer 1

Answer:

See below for answers and explanations

Step-by-step explanation:

Problem 1

$u\cdot v=\langle4,-7\rangle\cdot\langle-6,9\rangle=(4*-6)+(-7*9)=-24-63=-87$, so A is correct. Remember to multiply the horizontal components with each other and then add that to the product of the vertical components to get the dot product.

Problem 2

Two vectors are orthogonal if their dot product is 0, which also means that they must be perpendicular to each other, so we can check this:

$u\cdot v=(6*-24)+(-9*36)=-144-324=-468$

Since the dot product of the two vectors is 0, they aren't orthogonal. Therefore, they are parallel since $v=4u$, making A the best choice.

Problem 3

Recall that the angle between two vectors is $\theta=cos^{-1}(\frac{u\cdot v}{||u||*||v||})$ where $u\cdot v$ is the dot product of vectors $u$ and $v$ while $||u||$ and $||v||$ are the magnitudes of vectors $u$ and $v$ respectively. If we let $F_1=u$ and $F_2=v$, we can then continue on:

$u\cdot v=(190*128)+(160*-121)=24320+(-19360)=4960$

$||u||=\sqrt{190^2+160^2}=\sqrt{61700}=10\sqrt{617}$

$||v||=\sqrt{128^2+(-121)^2}=\sqrt{31025}=5\sqrt{1241}$

$\theta=cos^{-1}(\frac{u\cdot v}{||u||*||v||})\\\\\theta=cos^{-1}(\frac{4960}{10\sqrt{617}*5\sqrt{1241}})\\\\\theta\approx83.49^\circ\approx83^\circ$

Therefore, B is the correct answer

Problem 4

We observe that $u=\langle-8,-9\rangle$ and $v=\langle3,7\rangle$, so $-3(u\cdot v)=-3[(-8*3)+(-9*7)]=-3[-24+(-63)]=-3(-87)=261$. This means that B is the correct answer.

Problem 5

Recall that the projection of a vector $u$ onto $v$ is $proj_vu=(\frac{u\cdot v}{||v||^2})v$, thus:

$proj_vu=(\frac{u\cdot v}{||v||^2})v\\\\proj_vu=(\frac{(-10*4)+(-3*8)}{\sqrt{4^2+8^2}^2})\langle4,8\rangle\\\\proj_vu=(\frac{-40+(-24)}{\sqrt{80}^2})\langle4,8\rangle\\\\proj_vu=(\frac{-64}{80})\langle4,8\rangle\\\\proj_vu=(-\frac{4}{5})\langle4,8\rangle\\\\proj_vu=\langle-\frac{16}{5},-\frac{32}{5}\rangle$

Therefore, D is the correct answer