Calculus Vectors Q - NEED HELP ASAP!

[VCE1996]

Hi guys,

I need help with this question on vectors ASAP!

from a space shuttle Jupiter is in the direction of vector (1, -8, -4) and Mars in the direction (3, 12, 4). humans eyes have a field of view of 5pi/6 radians. determine if a human in space shuttle would be able to see jupiter and mars at the same time

Thanks a lot!

[hobbitle]

1) draw a picture (you dont really need it but I tend to do it on principle if the question is in 2d or 3d)
2) you will have been taught a way to work out the angle between two vectors. Have you covered cos a = v.w / ||v|| ||w||? Work out the angle between the two vectors.
3) is this angle more or less than 5pi/6 radians.

[VCE1996]

oh thanks so much! 

[hobbitle]

yw, hope it helped some.

[VCE1996]

one question.. if its more than 5pi/6 this means the alien cannot see both
if its less it means it can right?

[hobbitle]

Yes...

[VCE1996]

thanks again! 

do you think you could help me with this question too please?

Tasmania Jones, in his boat, needs to prevent a bomb from going off on an island 7km NE. The bomb goes off in 21 mins, but he arrives with 1 min to spare.

a) If the unit vector i represents a velocity of 1km/h east and j the unit vector with a velocity of 1km/h north, what is the vector representing Jone's required velocity?

If a jet ski has velocity b relative to the water it's in, and the water's flowing with velocity w relative to land, then the true velocity relative to land of the jet ski is b + w

b) after 5 mins Jones intercepts a current flowing due south at 30/sqrt2 km/h. What must he change his velocity relative to the water to?

c) Jones is unaware that his enemy has planted a bomb to his boat which will explode if the speed (magnitude of the velocity) of the boat (relative to the water) exceeds 40km/h. will jones survive?

[hobbitle]

I recommend you draw pictures, with these kind of vector questions.

Its difficult to visualise all of the information in your head in one go without doing it in steps.

In my explanation I'm going to assume you understand the concept of vector notation i.e. v = xi + yj where x is the units moved in the horizontal x direction and y is the units moved in the vertical y direction.  I've looked at my Calculus 1 notes from last year and so this should definitely be familiar to you.

I'm also going to assume you know your standard triangles.

It's perhaps a bit confusing because instead of dealing in lengths, they have made the 'units' km/h (instead of units of length).  But other than that, it's all the same as what you've been doing in class.

I'm not going to show you all the working, and your tutors will need to see it, so you can't just copy what I write here, you must try to understand it.

So, for a)

He travels 7km in 20 minutes as stated in the problem so therefore his necessary velocity is 21km/hr NE.  Presumably you realise that NE means that he is travelling at an angle of pi/4 to the horizontal... so break out your standard triangles.

To turn this into a vector you need to break it down into i (horizontal) and j (vertical) components, which are clearly going to be the same (I've called them a).
cos (pi/4) = a/21    (SOHCAHTOA)
a = 21/sqrt(2)

So now you know what his velocity vector will be... this is the form: v = ai + aj 

b)

So suddenly he gets hit with a current (c) of 30km/h due south.  In vector notation, this is c = -30j

We are told that his true velocity = velocity relative to water + the water flow velocity.
He needs to maintain his true velocity v = 21/sqrt(2)i + 21/sqrt(2)j and we know the water flow velocity to be c = -30j.  We need to find his new velocity relative to the water, v_new = ?i + ?j

So you substitute these values in and rearrange the equation so you end up with ?i + ?j = (51/sqrt(2))i + (21/sqrt(2))j and you can equate the coefficients.

Now you know the i and j components of his REQUIRED velocity vector once he hits the current.

c) You only know the i and j components of his velocity vector but you need to figure out his actual velocity.  Draw a picture and use pythagoras to figure out the 'hypotenuse' 'length' (I hesitate to say length because in this question you are finding velocity not length but anyway it's the same process).

You should be able to answer that he will be safe because his velocity only needs to increase to 39km/h.

[VCE1996]

thanks heaps!!

[VCE1996]

Hey,

for part a) he travels 7km in 20 mins so then why do we use 21 when solving for cos pi/4??

[hobbitle]

Hey,

for part a) he travels 7km in 20 mins so then why do we use 21 when solving for cos pi/4??

Because he's travelling at 21km/hr.
(60 minutes in an hour / 20 minutes) * 7 km/hr = 21 km/hr

[juzza12]

I recommend you utilise WOLFRAM ALPHA when undertaking Specialist Mathematics practice questions, this is an excellent resource and can be extremely beneficial to SAC results and time taken when completing excercises from the textbook (Essential Specialist Mathematics) of course.
Cheers

[hobbitle]

Agreed, although I think the OP is taking Calculus 1 at UoM but yeah same difference.