Vector component displacements problem

In summary, d1 - d2 + d3 = 20.3 degrees. The angle between r and the positive z axis is 133.82 degrees. The component of d1 along the direction of d2 is 20.3 degrees. The component of d1 that is perpendicular to the direction of d2 and in the plane of d1 and d2 is 3.3 degrees.
  • #1
mmg0789
39
0

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 don't know what phi is

thanks
 
Last edited:
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  • #2
mmg0789 said:
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 don't 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
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
mmg0789 said:
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
(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
Shooting star said:
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).

mmg0789 said:
(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
I think i understand
Thanks for your help!
 

Related to Vector component displacements problem

What is a vector component displacement problem?

A vector component displacement problem is a physics problem that involves finding the displacement of an object in two or three dimensions using vector components. It requires knowledge of vector addition and trigonometry to solve.

What are vector components?

Vector components are the individual parts of a vector that act in different directions. They are typically represented by a horizontal and vertical component, but can also be represented in three dimensions with x, y, and z components.

How do you find the resultant displacement using vector components?

To find the resultant displacement, you must first break down the vectors into their horizontal and vertical components. Then, use vector addition to find the sum of these components. The magnitude and direction of the resultant vector can be found using Pythagorean's theorem and trigonometric functions.

What is the difference between displacement and distance?

Displacement is a vector quantity that represents the change in position of an object from its initial to final position, taking into account direction. Distance, on the other hand, is a scalar quantity that measures the total length of the path traveled by an object, without considering direction.

Can vector component displacements be negative?

Yes, vector components can be negative. Negative components indicate a direction opposite to the positive direction on a coordinate axis. However, the magnitude of the vector component will always be positive.

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