Vector component displacements problem

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Homework Statement



Here are three displacements, each in meters: d1 = 3.3i + 3.8j - 8.2k, d2 = -1.0i + 2.0j + 3.0k, and d3 = 4.0i + 3.0j + 2.0k. What is r = d1 - d2 + d3 ((a), (b) and (c) for i, j and k components respectively)? (d) What is the angle between r and the positive z axis? (e) What is the component of d1 along the direction of d2? (f) What is the component of d1 that is perpendicular to the direction of d2 and in the plane of d1 and d2?

The Attempt at a Solution



a) 8.3i
b) 4.8j
c) -9.2k
d) arccos(-9.2/13.3) = 133.82 degrees

also here, i was just wondering: what if they wanted the angle between r and y axis? would that be arcsin(4.8/13.3)? would x axis be arccos(8.3/13.3)?

E and F are my problems

E sounds like the definiton of dot prouct, so:
d1 dot d2 = (3.3)(-1) + (3.8)(2) + (-8.2)(3) = 20.3
it seems right to me...but i think its wrong

F sounds like cross product, so:
c = d1*d2 sin(phi), i can figure out the mags of d1 and d2, but i dont know what phi is

thanks
 
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Answers and Replies

  • #2
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E and F are my problems

E sounds like the definiton of dot prouct, so:
d1 dot d2 = (3.3)(-1) + (3.8)(2) + (-8.2)(3) = 20.3
it seems right to me...but i think its wrong

F sounds like cross product, so:
c = d1*d2 sin(phi), i can figure out the mags of d1 and d2, but i dont know what phi is

thanks
For E, they want the component of d1 along the direction of d2. So, you should take the dot product of d1 with the unit vector along d2.

For F, if you take away the component of d1 along d2 form d1, what remains? Remember that a vector can be resolved into two mutually perpendicular components.
 
  • #3
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so for F it would be d1(z) - (d1 dot d2) ?

also i showed my calculations for E in my first post

d1 dot d2 = d1(x)*d2(x) + d1(y)*d2(y) + d1(z)*d2(z)
which would be (3.3)(-1) + (3.8)(2) + (-8.2)(3) = 20.3
would that be right?

thanks
 
  • #4
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so for F it would be d1(z) - (d1 dot d2) ?

also i showed my calculations for E in my first post
To find the unit vector along any vector A, you have to take the quantity A/mod(A). So , you have to divide the vector by its magnitude. [mod(A) means the magnitude.]

Let e denote the unit vector along d2 => e=d2/mod(d2).

So, the component of d1 along d2 is d1.e = d1.d2/mod(d2).

For F, this component has to be subtracted from the whole d1, not d1(z).
 
  • #5
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(3.3i + 3.8j - 8.2k) dot (1.0i + 2.0j + 3)((-1i + 2j +3k)/(3.74))

ahh ok ok its starting to make sense...find the unit vector of d2 to know where d2 points, then dot product that unit vector with d1 to point d1 in the direction of d2. right?
 
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  • #6
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Let e denote the unit vector along d2 => e=d2/mod(d2).

So, the component of d1 along d2 is d1.e
That is the magnitude of the component. To make it into a vector in the direction of e, it should be d1.d2/mod(d2)e = [d1.d2/mod(d2)]d2/mod(d2).

So, you have to divide twice by mod(d2).

(3.3i + 3.8j - 8.2k) dot (1.0i + 2.0j + 3)((-1i + 2j +3k)/(3.74))

ahh ok ok its starting to make sense...find the unit vector of d2 to know where d2 points, then dot product that unit vector with d1 to point d1 in the direction of d2. right?

Almost there.

[(3.3i + 3.8j - 8.2k) dot (1.0i + 2.0j + 3)((-1i + 2j +3k)/(3.74))]/(3.74)
= [(3.3i + 3.8j - 8.2k) dot (1.0i + 2.0j + 3)](-1i + 2j +3k)/13

should do it.

Subtract this from d1 to get the ans for (F).
 
  • #7
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I think i understand
Thanks for your help!
 

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