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math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Vectors

Hello;
While revising for an exam ( 2 days away ), I came across these questions. I have absolutely no idea how to go about the solutions. Please help.

(1)  Show that the points with position vectors 4i + 5j, 3i + 3j and -3j are collinear.

(2)  A body is moving with velocity v ms^(-1) where v = 2i - 3j. If it started from the position i + 4j, what is its position after 3 seconds? How long will it take to reach the position 11(i - j)?

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math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Vectors

(3)  Find the gradient and equation of the line given by r = i - j + k(i - j) where k is a number.

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math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Vectors

(4) If a = 3i + 4j and |b| = 2, what are the greatest and smallest values of |a + b|?

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math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Vectors

Question 5

(a)  The velocity v ms^(-1) of a body is given by the vector v = i + 3j. Find the speed of the body and the angle its path makes with the x-axis.
(b) If its position vector at the start was i + j,  what is its position vector (i) after 1 sec, (ii) after 3 secs, (iii) after t secs ?
(c) After what time will it reach the position given by 7i + 19j?

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math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Vectors

I don't know how to use vectors to find speed and velocity so this is all new to me. Sorry I couldn't type the vectors in bold print. Hope you can make them out though.

The book provides answers so we can also compare.

Many thanks in advance.

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zetafunc

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Registered: 2014-05-21

Posts: 2,436

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Re: Vectors

For (1), you've got to show that there exists a straight line passing through each of those points. Try looking at the difference between 4i + 5j and 3i + 3j.

For (2), the body is moving 2i - 3j metres every second, i.e. it moves 4i - 6j after 2 seconds, 6i - 9j metres after 3 seconds, and so on.

For (3), since this is a two-dimensional problem, you might find it helpful to simply plot i - j in Cartesian co-ordinates.

For (4), what are the possible values of b?

For (5) (a), speed is the magnitude of the velocity vector. So what's the magnitude of i + 3j?

For (5) (b), apply what you've done in (2).

Ditto for (5) (c).

Last edited by zetafunc (2015-07-21 01:12:43)

math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Vectors

Hi zetafunc,

Thanks for reply.

For (4), the question gives no value(s) of b. It's just as I typed it.

For (1), after finding the difference between those two, then. . . .? Does it have to do with the gradients being the same?

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Vectors

Q1.  For points A, B, and C, find the vector AB and the vector BC.

If AB = scalar x BC, then you know they are parallel, and as they have B in common, they must all be in one line.

Q2.  The equation of the line of travel is

r = i + 4j + t(2i - 3j)

The parameter t will be the time in seconds so you can put t = 3 for the first part and solve for t in the second.

Q3. When k = 0, a point on the line is i - j.

Choose any other value for k and you'll get a second point.

then the gradient = (y2-y1)/(x2-x1) and substitution into y = mx + c will give you c.

Q4.  The locus of points for b, is a circle (one) radius 2, centred on (0,0).

So if you add another 3i+4j to this you'll create a new circle (two) translated by that amount.  If you draw a diagram the greatest and least should be obvious.  (I think.  I'm going to try this myself. )

OK.  Done that now.  Draw and extend the line from (0,0) to (3,4) and mark the two points where it cuts circle two.  Any point on circle two is a possible a + b, but the two points you've just marked will give the greatest and least distances from (0,0) ie. will maximise and minimise |a+b|

Bob

note: If distances are in say metres, velocities in m/s, and time in seconds then distance travelled d = t.v.

And speed = |v|

And you can make bold with square brackets b to turn on and square brackets /b to turn off.

Last edited by Bob (2015-07-21 03:05:25)

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math9maniac

Member

From: Tema

Registered: 2015-03-30

Posts: 443

Re: Vectors

Thanks Bob and zetafunc. I've been able to solve 1, 3 and 5. Will be going to bed soon so I'll check others tomorrow. Since there's no paper tomorrow, I'm spending the day at home although inside me I'd love to be by the beach. It's so serene over there and conducive for learning. Well too bad.

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