MHB Reviewing Cross Product: Simplest Method Possible

[ineedhelpnow]

HEY GUYS! (Wave)
ok so i have this question i did. and now I am reviewing for the test and i looked at how i did it and i did in the most complicated way ever. i don't FULLY understand chegg's method. so i hope someone can provide me with the SIMPLEST method possible. thank u! (Blush) (p.s. don't ask me what the green stain on my final answer is because i have no idea) i would really appreciate a step by step solution at one shot so i can fully understand it.

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$\left| \tau\right| = \left| r \right| \left| F \right| sin (\theta)$

 

[I like Serena]

HEY GUYS! (Wave)
ok so i have this question i did. and now I am reviewing for the test and i looked at how i did it and i did in the most complicated way ever. i don't FULLY understand chegg's method. so i hope someone can provide me with the SIMPLEST method possible. thank u! (Blush) (p.s. don't ask me what the green stain on my final answer is because i have no idea) i would really appreciate a step by step solution at one shot so i can fully understand it.

Your result is correct! (Yes)

Here is how I would do it.

The formula for torque is:
$$\boldsymbol{\tau} = \mathbf r \times \mathbf F \tag 1$$

Normalizing the direction vector of the force, we have:
$$\mathbf{\hat F} = \frac 1 5 \begin{pmatrix}0\\3\\-4\end{pmatrix} \tag 2$$

Substituting the data of $\mathbf r$ and $(2)$ into $(1)$ we get:
$$\boldsymbol\tau
= \begin{pmatrix}0\\0.3\\0\end{pmatrix}
\times \frac 1 5 F \begin{pmatrix}0\\3\\-4\end{pmatrix}
=\frac{1}{50}F \begin{pmatrix}0\\3\\0\end{pmatrix}
\times \begin{pmatrix}0\\3\\-4\end{pmatrix}
=\frac{1}{50}F \begin{pmatrix}-12\\0\\0\end{pmatrix}
$$

So:
$$\tau=\frac{12}{50}F=100 \Rightarrow F = 417 \text{ N}$$

Btw, what is that $\color{Green}{\text{green stain}}$ on your answer?
Were you eating? (Wondering)

 

[ineedhelpnow]

ive never really seen it don't that way. is there an easy way to first solve for theta and then continue the rest of the problem.

lol i don't know. i think its a food stain. i was eating while doing it. i think its like oil or something (Giggle)

 

[ineedhelpnow]

now that i look back at it, i think it does make sense the way i did it. when i was doing it, i didnt get it much but now i realize that all i did was use the dot product as well as the cross product. i think i understand it more now because I am familiar with the equation $r \cdot v= \left| r \right| \left| v \right| cos (\theta)$

 

[I like Serena]

ive never really seen it don't that way. is there an easy way to first solve for theta and then continue the rest of the problem.

That's how your solution works.
First use the dot product to find the angle between the 2 vectors:
$$\mathbf r \cdot \mathbf v = r_x v_x + r_y v_y + r_z v_z$$
$$\mathbf r \cdot \mathbf v = |r|\cdot |v|\cdot \cos(\theta)$$
Then use the cross product to find the torque:
$$\tau = |\mathbf r \times \mathbf F| = |r|\cdot |F|\cdot \sin(\theta)$$

In my version I skipped those steps with the dot product, and went straight for the cross product.