How can I find the component of a cross product along a given vector direction?

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Homework Help Overview

The discussion revolves around finding the component of a cross product along a specified vector direction, specifically using vectors from a physics context. The original poster references a problem from a textbook involving vectors A and B, and their cross product D, along with a direction vector C.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to calculate the component of the cross product D along vector C using angle and magnitude, while another participant suggests using the dot product with a unit vector instead. Additionally, a separate problem involving hyperbolas and gradients is introduced, where the poster considers finding the component of vector A in the direction of the gradient of u.

Discussion Status

Participants are exploring different methods to find vector components, with some guidance provided on using the dot product with unit vectors. There is an acknowledgment of complexity in the original poster's approach, and a shift to a new problem indicates ongoing exploration of related concepts.

Contextual Notes

The original poster expresses confusion regarding the correctness of their calculations, noting discrepancies with textbook answers. The second post questions the necessity of using angles in the calculations, suggesting a more straightforward method. The discussion also touches on the gradient of a function and its application to vector components.

Noone1982
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I'm using wangsness EM book. I keep doing ch 1 #5 and keep getting 19, despite the back of the book saying -14.4

A = <2,3,-4>
B = <-6,-4,1>

AxB = <-13, -22, 10> = D

Find the component of AXB along the direction of Vector C

C = <1,-1,1>

Now I am using

CD = CDcos(theta)

Im getting 66.4º

Now, to get the component I used:

Magnitude = DCcos(66.4)

Does this method make sense?
 
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To get the component of D in the direction of C, you need to take the dot product of D with the unit vector in the direction of C. (Divide [itex]\vec{D} \bullet \Vec{C}[/itex] by the magnitude of C.)

(Also: Why mess around with angles when you have the components?)
 
Thanks for the help, I tend to do things the hard way. I have another question:

"A family of hyperbolas in the xy plane is given by u = xy. Find the gradient of u. Given vector A = 3i + 2j + 4k find the component of A in the direction of gradient u at the point u = 3 and x = 2"

Since we know that u = 3 and x = 2, we can gather that the point of the hyperbola is x =2 and y = 3/2

Lets call this vector E

so E = 2i + (3/2)j + 0k

This problem is similar to the previous one I asked, finding the component. So I take it since I want to find the component of A in the direction of Grad U, I will need to make E into a unit vector

U = (4/5)i + (3/5)j + 0k

Now I presume I dot U and A?

That would create

UA = (8/5) +(9/10) = 5/2
 
Ignore the previous post. I was experiencing brain drain...
 

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