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Vector Problems
Started by Loren8443
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I was searching online for the PCAT Practice test, and I came across this one website, WWW.THEPCAT.COM, I wana to know if you are familiar with this website. I would like to know if this is a helpful website or not. check this website out. No, i am so sorry i dont know any thing about Vector. Let me know if you know anything about this.
Thanks a bunch
I haven't seen this site before. It seems like it'd be good to check out, although if you're taking the August PCAT, it might not be worth it. I've been studying from Kaplan, ARCO, and Barrons, and I ordered Cliffs and am waiting for it to arrive. I just want to make sure I have all the bases covered, and am still trying to figure out vectors.
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OK, well I know nothing about what the PCAT tests (pre-med here), but I do know a good amount of linear algebra.
An easy way to think about vectors in 2 dimensions (which is probably all you'll need to know) is to imagine arrows. Let's say arrow a is 4 inches long, and arrow b is 3 inches long.
----> (Arrow a)
<--- (Arrow b)
So, now let's add a + b. (Note: Vector's are usually represented by lowercase letters with little arrows over them, or sometimes just bolded lowercase letters)
----> + <---
I'm sure at this point you can guess intuitively what will happen here. The two arrows kind of cancel each other out, right? It's like you traveled towards the right 4 inches, and then decided to go left 3 inches. What does that give you? Well, it means you've displaced yourself 1 inch to the right of your original position.
Vectors are used when dealing with displacement in physics, which is how far you've moved from your initial position. If you wanted the distance (which is a scalar, NOT a vector) you would say you've moved a total of 7 inches (4 to the right, 3 to the left). So if you ran around a track for 4 laps, your distance travelled would be 1 mile, but your displacement is zero because you're at the same point you started at!
Vectors obviously don't just add in simple left-to-right, up-and-down directions. They can go diagonally as well. If this is the case, use the "tip-to-tail" rule. You put the tip of the first arrow you have at the "tail," or end, of your second arrow. Now, draw a hypotenuse and use a^2 + b^2 = c^2 to find out the displacement vector (represented here by c).
Ex:
a = 3 inches b=6 inches
^ ------>
|
|
|
The bolded /'s represent c.
----->
^ /
| /
| /
|/
3^2 + 6^2 = c^2
9 + 36 = c^2
sqrt(45) = c = 6.708
Sorry if this is a lot to absorb, and I don't have a scanner. Google tip-to-tail, though, I'm sure there are plenty of good examples.
If you want more mathematical definitions, consider the definition of a vector:
1) (u+v)+w = u+(v+w)
2) u + v = v + u
3) v + 0 = v
4) v + (-v) = 0
5) b(u + v) = bu + bv
6) (a+b)u = au + bu
7) a(bv) = ab(v)
8) 1u = u
The above 8 rules are really all you need to know as far as basic vector addition.
As you can see from Rules 5,6, and 8, multiplication between a scalar and a vector is easy; it can increase the magnitude (length, here) of a vector or shrink it. So if a is 6 inches
------>
Then 5a is 30 inches long in the same direction:
------------------------------>
Mulitiplication of two vectors are called a dot product. Basically, you just multiply the transpose of the first times the second. I don't think I can use sigma notation on SDN, so I'll just refer you here for an easier to read explanation of what a transpose is (it's basically just flipping the elements of a vector or matrix from horizontal to vertical arrangements)
http://en.wikipedia.org/wiki/Dot_product
Shameless plug: If you're ever interested, you can read up more on linear algebra; I wrote some chapters on the subject at Wikibooks, feel free to peruse
http://en.wikibooks.org/wiki/User:%C3%89variste
An easy way to think about vectors in 2 dimensions (which is probably all you'll need to know) is to imagine arrows. Let's say arrow a is 4 inches long, and arrow b is 3 inches long.
----> (Arrow a)
<--- (Arrow b)
So, now let's add a + b. (Note: Vector's are usually represented by lowercase letters with little arrows over them, or sometimes just bolded lowercase letters)
----> + <---
I'm sure at this point you can guess intuitively what will happen here. The two arrows kind of cancel each other out, right? It's like you traveled towards the right 4 inches, and then decided to go left 3 inches. What does that give you? Well, it means you've displaced yourself 1 inch to the right of your original position.
Vectors are used when dealing with displacement in physics, which is how far you've moved from your initial position. If you wanted the distance (which is a scalar, NOT a vector) you would say you've moved a total of 7 inches (4 to the right, 3 to the left). So if you ran around a track for 4 laps, your distance travelled would be 1 mile, but your displacement is zero because you're at the same point you started at!
Vectors obviously don't just add in simple left-to-right, up-and-down directions. They can go diagonally as well. If this is the case, use the "tip-to-tail" rule. You put the tip of the first arrow you have at the "tail," or end, of your second arrow. Now, draw a hypotenuse and use a^2 + b^2 = c^2 to find out the displacement vector (represented here by c).
Ex:
a = 3 inches b=6 inches
^ ------>
|
|
|
The bolded /'s represent c.
----->
^ /
| /
| /
|/
3^2 + 6^2 = c^2
9 + 36 = c^2
sqrt(45) = c = 6.708
Sorry if this is a lot to absorb, and I don't have a scanner. Google tip-to-tail, though, I'm sure there are plenty of good examples.
If you want more mathematical definitions, consider the definition of a vector:
1) (u+v)+w = u+(v+w)
2) u + v = v + u
3) v + 0 = v
4) v + (-v) = 0
5) b(u + v) = bu + bv
6) (a+b)u = au + bu
7) a(bv) = ab(v)
8) 1u = u
The above 8 rules are really all you need to know as far as basic vector addition.
As you can see from Rules 5,6, and 8, multiplication between a scalar and a vector is easy; it can increase the magnitude (length, here) of a vector or shrink it. So if a is 6 inches
------>
Then 5a is 30 inches long in the same direction:
------------------------------>
Mulitiplication of two vectors are called a dot product. Basically, you just multiply the transpose of the first times the second. I don't think I can use sigma notation on SDN, so I'll just refer you here for an easier to read explanation of what a transpose is (it's basically just flipping the elements of a vector or matrix from horizontal to vertical arrangements)
http://en.wikipedia.org/wiki/Dot_product
Shameless plug: If you're ever interested, you can read up more on linear algebra; I wrote some chapters on the subject at Wikibooks, feel free to peruse
http://en.wikibooks.org/wiki/User:%C3%89variste
To those who took the exam already, what sort of vector problems should we expect? Hopefully they will mostly be 1D or 2D.
And as the OP mentioned, if anyone knows of any sites with some practice problems, mucho thanks...
And as the OP mentioned, if anyone knows of any sites with some practice problems, mucho thanks...
thanks for the overview on vectors! i remembered doing them in precalc, linear algebra, and 1 of the 4 calculus classes... but oh well, no matter how many times i take these classes, the material doesnt absorb! so thanks again!
You know what I was thinking...What if on the Aug PCAT they don't give us vector problems. I mean Harcourt is not stupid, they know people will tell others that there were vector problems. What if they surprise us with something else that we hadn't thought of, just like vectors was a surprise for the June PCAT takers. What do you guys think?
Thank you for those links, Mr. Sexcrime.
Anyway, final stretch for the August heads! Good luck! Still super-stressed about the chemistry part........
Anyway, final stretch for the August heads! Good luck! Still super-stressed about the chemistry part........
both links work...thanks.
ps: carl is the man!!!!!
ps: carl is the man!!!!!
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