Help with vectors

**Laythen** · Oct 21st 2011, 09:05 AM

Hello, I need help with my introduction to vectors book. I am trying to find out how I can draw vectors knowing only their norms and angles. How long do I draw them? How do I figure out the coordinates of the tail and the head from only knowing the norm? I know the tail will be (0,0) but I need to know how to figure out the coordinates to the head.

Here is a question scanned from my book.

"Given the norms and measures of the angles of direction U and V, find the following linear combinations in order to determine the norm and the measure of the angle of direction of W."

Any help appreciated please, I have been trying to figure this out all day and it's really the only thing stumping me so far in the book. Thank you

**Duke** · Oct 21st 2011, 09:15 AM

A vector's length and direction does not determine their position in space.

**Duke** · Oct 21st 2011, 09:16 AM

They are talking about position vectors i.e start at the origin.

**Laythen** · Oct 21st 2011, 09:27 AM

Thanks for the reply but I'm not sure what you mean. I just started the book and have no previous experience with them. I will scan and upload the answer. Notice -->4V starts at (0,0) and has a head of (-8,-12)

Why does the head end there? How can I get that information from simply having the norm and the angle. I need to know how long to draw the vectors to get the middle one (-->W)

**Duke** · Oct 21st 2011, 10:42 AM

let v=(v1,v2). Then |v|= $sqrt((v_1)^2+(v_2)^2)$ and $tanx=v_2/v_1$

**Laythen** · Oct 21st 2011, 11:33 AM

I feel so silly but I still don't understand. I don't have any of the coordinates for vector V or vector U. I need to figure out how to get those. I can use that equation to get the norm for V but I have it already. I need to get the coordinates of the head of V from the norm and its angle.

I need to know how to get (-8,-12) which is the head of vector V. Given that vector V is 4v and norm is //v// = 3.6. Then, if I know that, I'll be able to use the same formula/way to get the head of U and any other vectors for other problems.

**HallsofIvy** · Oct 21st 2011, 01:00 PM

If two dimensional vector, v, has norm (length) r and makes angle $\theta$ with the positive x-axis its x component is $r cos(\theta)$ and its y component is $r sin(\theta)$ .

However if you are given vectors in terms of "norm and direction" and are expected to give the answer in the same way, then it makes sense to use the cosine law and sine law. 2u has length 6.4 and makes angle with 342 degrees with the x-axis (that is my interpretation of "angle of direction") while 4v has length 14.4 and makes angle with 236 degrees with the x-axis. If you "move" the tail of v to the head of v, then connect the head v to the tail of u you get a triangle with two sides of length 6.4 and 14.4 and with angle at the point where the two vectors "join" of 74 degrees. The cosine law will give you the length of the third side, ||2u+ 4v|| and sine law will give you the other angles from which you can calculate the angle this last vector makes with the x-axis- its "angle of direction".

**Laythen** · Oct 21st 2011, 01:24 PM

Originally Posted by HallsofIvy

If two dimensional vector, v, has norm (length) r and makes angle $\theta$ with the positive x-axis its x component is $r cos(\theta)$ and its y component is $r sin(\theta)$ .

EXACTLY what I needed to know. Thank you very much, it checks out for both vector U and V (finds the equation of the head.)

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