Question

I could really use some help with these 4 problems. Any help will be greatly appreciated!

Answer 1

(7) is the same problem with different numbers as (4) in another question you posted: https://brainly.com/question/11799042

(8) is not correct. $\mathbf u\cdot\mathbf v=(12)(-15)+(4)(-5)=-200$

(9) is not correct. The direction of the vector $\langle3,4\rangle$ is $\theta$ where

$\tan\theta=\dfrac34\implies\theta=\tan^{-1}\dfrac34$

so the applied force is the vector

$\mathbf F=\langle14\cos\theta,14\sin\theta\rangle=\langle11.2,8.4\rangle$

(if you're not sure how to compute the (co)sine of the inverse-tangent expression, note that a tangent of 3/4 corresponds to a right triangle with hypotenuse 5)

The object is moved 7 ft in the same direction as $\mathbf F$, so that its displacement is given by the vector

$\mathbf d=\langle7\cos\theta,7\sin\theta\rangle=\langle5.6,4.2\rangle$

Then the work done on the object is the dot product of these two vectors,

$W=\mathbf F\cdot\mathbf d=(11.2)(5.6)+(8.4)(4.2)=98\approx97$

(the discrepancy here might be coming from finding a numerical value for $\theta$)

(10) is correct. We can find a unit vector pointing in the direction of any vector $\mathbf v$ by dividing $\mathbf v$ by its magnitude:

$\dfrac{\mathbf v}{\|\mathbf v\|}=\dfrac{\frac23\,\mathbf i-\frac12\,\mathbf j}{\sqrt{\left(\frac23\right)^2+\left(-\frac12\right)^2}}=\dfrac45\,\mathbf i-\dfrac35\,\mathbf j$