# Homework Help: Physics 221 Calc-based question for homework

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1. Jan 20, 2017

### AidenPhysica

1. The problem statement, all variables and given/known data
If vector d1--> is equal to 2i^-4j^+9k^ and vector d2-> is equal to 8i^-10j^+9k^, what is (d1-->+d2-->) * (d1--> X 4d2->)?

2. Relevant equations

3. The attempt at a solution
So I added up first two vectors and then multiplied it by the cross product but I get a vector and when I find its length its like 1931. I just want to know if this is even close.

2. Jan 20, 2017

### Staff: Mentor

Hi AidenPhysica, Welcome to Physics Forums.

Your result does not look correct to me. Can you show the details of your calculations?

3. Jan 20, 2017

### Staff: Mentor

$$(\vec{d_1} + \vec{d_2}) \cdot (\vec{d_1} \times 4 \vec{d_2})$$
?

If that is correct, then your result is wrong. What to you get when you take the cross product of two vectors?

4. Jan 20, 2017

### AidenPhysica

Its right what you put except there has to be a 4 right in front of d2 so its multiplied by four.

5. Jan 20, 2017

### Staff: Mentor

I've corrected it. So, what do you get when you take the cross product of two vectors?

6. Jan 20, 2017

### AidenPhysica

Ok, so this is what I did. I first added up the two vectors to get 10i+-14j+18k. Then you do the cross product, so I did that operation of the cross product, which is basically how much two vectors are perpendicular to each other? So here is the matrix i j k on top second row is 2, -4, 9. Third row is 32i-40j+36k. The cross product of this matrix is

7. Jan 20, 2017

### AidenPhysica

216i +216j+48k

8. Jan 20, 2017

### Staff: Mentor

I'll give you a hint: the actual values of the vectors are not important.

What is the orientation of that vector with respect to the original vectors?

9. Jan 20, 2017

### AidenPhysica

Then what I did was I multiplied each respective component of the first answer times the second to get 3815 after doing pythagrean theorem is that better?

10. Jan 20, 2017

### AidenPhysica

OK, so what do you mean by orientation?

11. Jan 20, 2017

### Staff: Mentor

Stop a minute. I want you to think about the problem first. You can calculate numerical values later.

The direction in which it points. When you take the cross product of two vectors, the resulting vector points in a very specific direction with respect to the two other vectors.

12. Jan 20, 2017

### AidenPhysica

so isn't the cross product equal to magnitude of a times magnitude of b multiplied by sin of theta?

13. Jan 20, 2017

### AidenPhysica

Isn't cross product equal to magnitude of a times magnitude of b times sin of theta?

14. Jan 20, 2017

### AidenPhysica

so isn't it just: ok so we have 10.05 times 26.83 times sin of theta=309.22. So then theta=309.22/269.6*(arcsin)

15. Jan 20, 2017

### AidenPhysica

I don't know it doesn't work.

16. Jan 20, 2017

### AidenPhysica

I try plugging it into a times b sin theta = magnitude of cross product but it just doesn't work.

17. Jan 20, 2017

### Staff: Mentor

18. Jan 20, 2017

### AidenPhysica

so how do i find orientation though?

19. Jan 20, 2017

### AidenPhysica

Is it in degrees? I am just confused I see the picture, its like perpendicular.

20. Jan 20, 2017

### Staff: Mentor

Yes, the resulting vector is perpendicular to the plane of the two initial vectors.

21. Jan 21, 2017

### AidenPhysica

I'm sorry can you please tell me at least the next step, i am confused and have been at this for hours.

22. Jan 21, 2017

### Ray Vickson

(1) What is the general formula for $\mathbf{c}= \mathbf{a} \times \mathbf{b},$ where $\mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}$ and $\mathbf{b} = b_x \mathbf{i} + b_y \mathbf{j} + b_z \mathbf{k}?$ Express your answer as $\mathbf{c} = c_x \mathbf{i} + c_y \mathbf{j} + c_z \mathbf{k},$ and give explicit formulas for $c_x, c_y, c_z$ in terms of the $a$'s and $b$'s. If you do not know the formula, you can Google "vector cross product" to find hundreds of articles that will be helpful.
(2) What is the geometric interpetation of $\mathbf{c}$ in relation to $\mathbf{a}$ and $\mathbf{b}?$
(3) How can you do the question without doing any calculations at all?

23. Jan 22, 2017

### AidenPhysica

Thank you all for the help, thank you Ray Vickson and the mentors as well. So Mr. Vickson, I found the answer to be 0 via calculations but I only vaguely understand why it is 0. Is it because whenever you take dot product, which is a scalar of how much one vector lies along another vector, and take the dot product within the cross product of those same vectors, which is the vector perpendicular to the two vectors, you cancel the two values out and it just leaves you with 0?

24. Jan 22, 2017

### Ray Vickson

Use the elementary fact that
$$(\mathbf{d}_1 + \mathbf{d}_2) \cdot \mathbf{c} = \mathbf{d}_1 \cdot \mathbf{c} +\mathbf{d}_2 \cdot \mathbf{c}$$
Here, $\mathbf{c} = \mathbf{d}_1 \times 4 \mathbf{d}_2 =4 ( \mathbf{d}_1 \times \mathbf{d}_2 ).$

Can you see why $\mathbf{d}_1 \cdot \mathbf{c}$ and $\mathbf{d}_2 \cdot \mathbf{c}$ are both equal to 0? (Hint: question number (2) in my response.)

Last edited: Jan 22, 2017