Calculating the magnitude of two vectors

[UnD3R0aTh]

they all have the same magnitude correct?

 

[arildno]

ok i tried the same method to verify the last statement in the problem, i calculated the magnitude of both vector D and E, i multiplied vector D by 2, i drew a triangle and i used the law of cosines, the lower bound is 16.9, the upper bound is 37.5, but the problem says that the value can never be higher than 16.9 but it might be smaller, what have i done wrong? (problem in the attachments)

Now, there are some important differences in this problem relative to the last one:
1. The components are GIVEN. That means the angle between the two vectors are necessarily fixed, so you do not have a range of thetas to look at.
2. But: The problem, precisely because you have the components (you didn't have that in the previous problem!) is easily solved:
Just COMPUTE vector 2D-E, and then determine its magnitude with the Pythagorean theorem.

 

[arildno]

they all have the same magnitude correct?

Yes.

 

[UnD3R0aTh]

Now, there are some important differences in this problem relative to the last one:
1. The components are GIVEN. That means the angle between the two vectors are necessarily fixed, so you do not have a range of thetas to look at.
2. But: The problem, precisely because you have the components (you didn't have that in the previous problem!) is easily solved:
Just COMPUTE vector 2D-E, and then determine its magnitude with the Pythagorean theorem.

did u read the last statement about the evaluation? what does it mean?

 

[arildno]

Yes.
It is a very nebulous statement, but it is critical to highlight something said previously:
"recall from Section 1.7 that subtracting a vector is the same as ADDING the negative of that vector"

Now, keeping THAT statement in mind, look at what "Evaluate" actually says:
"Our answer is in the same order of magnitude as the larger components that appear in that SUM"

That means, the components in SUM 2D+(-E) (prior to summing them, we have components 12, -4, 6, 5, -2, -8)

The biggest component here is the number "12", agreed?
-------------------------
So, the book is simply saying that when we subtract two vectors A and B from each other, we can look at the biggest components in A and B and say that the magnitude of the difference vector might be roughly as large as the biggest components in A or B (but never too much beyond that), but that magnitude CAN be much smaller (If A is a HUGE vector, but B=A, then A-B has zero magnitude!)

 

[arildno]

When we, however, look at the law of cosines for vectors of fixed magnitudes, we basically can estimate the maximal magnitude of the difference vector IF we allow the components to change (that's different from the present problem!)

In THAT case, we have found out that the maximal magnitude of the difference vector is the sum of the magnitudes of the two other vectors.

 

[UnD3R0aTh]

ok so u are saying that the biggest components in the sum are the i components which are 12 and -4 right? and the sum of those are 16, so we can't expect our answer to be larger than that right? but i don't understand this part "but that magnitude CAN be much smaller (If A is a HUGE vector, but B=A, then A-B has zero magnitude!)" how can it be much smaller?!

 

[mfb]

Consider the sum of two large vectors P=(-100,0,0) and Q=(101,0,0). They have a magnitude of 100 and 101, respectively. Their sum is R=(1,0,0) with magnitude 1, much smaller than the magnitudes of P and Q.

 

[arildno]

Well, i'd say that 6,5 and |-8| are components bigger than |-4|, don't you agree?

Isn't 0 the smallest non-negative number you can have?
Suppose a vector has component form 100*i
Then, if you subtract it from itself, the resultant vector has magnitude 0, quite a lot below 100.

 

[UnD3R0aTh]

Well, i'd say that 6,5 and |-8| are components bigger than |-4|, don't you agree?

Isn't 0 the smallest non-negative number you can have?
Suppose a vector has component form 100*i
Then, if you subtract it from itself, the resultant vector has magnitude 0, quite a lot below 100.

the biggest components in a specific direction, which are the closest to 16.9 are the components in the x direction of the two vectors, can i really add 12 and 8, although the 12 is the x-axis direction and the 8 is in the z axis direction?

 

[arildno]

the biggest components in a specific direction, which are the closest to 16.9 are the components in the x direction of the two vectors, can i really add 12 and 8, although the 12 is the x-axis direction and the 8 is in the z axis direction?

Of course not, and the book doesn't say that!
Rather, it says that we can make an easy, swift estimate of what the magnitude of the resultant vector maximally can be, if we just look at the magnitudes of our components.

So, in your case, prior to performing the sum and computing the magnitude, the book says you could think like this:
"Aah! my biggest component is 12, so I shouldn't expect my answer of magnitude to be a lot bigger than that!"
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If you think of it, to develop such rules as to what you might EXPECT answers to be can be a very useful control in order to gauge whether the ACTUAL answer you compute after messy calculations can actually be right.

It is so easy to make a slight mistake in a series of calculations, so every nice rule of thumb that can tell us what we might expect as our answer will, in time, improve our ability to correct our own faults.

 

[UnD3R0aTh]

ok, i will bother u with one more question, is it possible to determine the direction of the vector from those x,y, and z components? (i already know how to determine its magnitude), and how do u explain that when i substituted theta with 0, i got an answer exactly = the solution, that must mean that those two vectors have zero angle between them (hence my earlier question of how to determine the direction)

 

[arildno]

"is it possible to determine the direction of the vector from those x,y, and z components?"

Certainly. But here is the critical issue: Direction relative to WHAT?

If you think of 2-D vectors, we are so used to think of "direction" to mean "angle vector makes with positive x-axis"

There will be one and only one, vector who has the same direction (angle to x-axis) and magnitude (if you specify angles to be distinct numbers between 0 and 360).

But, what happens if we use THAT definition of "direction" (angle to positive x-axis) when we enter into full, 3-D space?

THAT definition of "direction" does NOT specify a single vector, but rather, a CONE of vectors with the vertex at the origin, the x-axis as its axis of symmetry, and where the opening angle of the cone (when we look at its cross section in the xy-plane) is twice the opening angle.

So, the upshot is that the "direction" concept is a bit trickier in 3-D than in 2-D.

 

[UnD3R0aTh]

u forgot this part "how do u explain that when i substituted theta with 0, i got an answer exactly = the solution, that must mean that those two vectors have zero angle between them"

 

[arildno]

u forgot this part "how do u explain that when i substituted theta with 0, i got an answer exactly = the solution, that must mean that those two vectors have zero angle between them"

THAT must be the result of a curious miscalculation you made utilizing the law of cosines!
Why?
Consider the following fact:
IF the angle between two vectors is 0 (that is, parallell), then the magnitude of the difference vector must simply be (largest magnitude minus smallest magnitude). (If angle is 180 degrees, the magnitude of the difference vector is the SUM of the two magnitudes!)

But, the magnitude of the vector 2D in your case is sqrt(184), whereas the magnitude of vector E is square root (105)

When either using the law of cosines correctly, or subtracting those two numbers from each other, you'll get 3.3 as minimum value, not 16.9

 

[arildno]

While I cannot second guess your previous calculation, I have a strong suspicion that you inserted cos(theta)=0, rather than (theta)=0. THat gives analytically the result 17 meters for the magintude

 

[arildno]

The angle between 2D and -E is almost, but not quite, 90 degrees (approx. 89.2), so such a micalculation (cos(theta)=0 implies 90 degrees) explains the correspondence of values.

 

[codeia]

In general, it does.
Why should it be close to the sum?

Did you consider some examples, like
S=(0,1), T=(1,0)
S=(5,2), T=(4,2)
S=(5,0), T=(-4,0)
...? This will help to get a feeling for the problem.

Why those specific coordinates?

 

[mfb]

This thread is over two years old.

The numbers are arbitrary, but I chose them in such a way that some relevant cases are covered.