Leah here from leah4sci.com/MCAT and in this video, I'll take you through the process of adding and
subtracting Vectors on the MCAT. In the last video we talked about Vectors and Vectors
components but an MCAT question may come up with a given more than 1 Vector and you have
to put them together. You can find my entire series on MCAT Physics
Translational Motion by visiting my website http://leah4sci.com/MCATPHYSICS. Now this
sounds like a very complicated topic, unless you understand what's going on and understand
how to break it up. A Vector's simply a line that goes from
A to B and if you have more than 1 Vector, imagine that you go from A to B and then you're
travelling from B to C. For example, imagine that this diagram represents a City Grid where
we have multiple blocks each at a length of 10 meters.
If I start at point A and walk to point C, I walked a total of 20 meters because I covered
the distance of two city block, that's my one Vector. However, what if I start at Point
D, walk for 10 meters and stopped at point E and then walk 10 more meters and stop at
point F? What I really did was walk a total distance from D through F for 20 meters.
So we took one Vector and we added it to the second Vector to get a total or Resultant
Vector. When you're adding Vectors that are parallel to each other or going in the
same direction, you simply add the values. So in the example of going from D through
F we had Vector D through E plus Vector E through F, that's 10 meters plus 10 meters
for a total distance of 20 meters. This is called the head to tail method but
students often confused what's going on. So simply think of the arrow this way: A Vector
is simply an arrow, where the tail of the arrow is where you start and the head of the
arrow is where you wind up. If you have one Vector, meaning you stop you
have that arrow head and then you start again, it only makes logical sense to start again
from where you ended and that's your new tail and once again when you stop, that�s
your new head. That's the idea of the head to tail method because the second tail immediately
follows that first head. If you see positives and negatives, that can
sometimes get confusing. I've heard a thought that a positive Vector that you put head to
tail but the negative Vector is tail to tail or head to head, I personally find that confusing.
Don't think of your Science as a way to turn your arrow instead, think of it as a
direction. In referring to this grid, we can say that
the negative X direction is left. The positive X direction is to the right. Going North that�s
the positive Y direction and South is the Negative Y direction.
So if I start out at point A and have my first Vector 20 meters to the right at point B and
then I stop at point B and change directions going 10 meters to the left, then yes, I wind
up having positive 20 because I walked to the right, negative 10 because I walked to
the left and that gives me a Resultant Vector of Twenty minus Ten equals ten meters (20-10=10m).
The idea of drawing this head to head to me is personally confusing, so instead, I have
my first Vector 20 meters to the right, my second Vector, 10 meters to the left and yes,
my Resultant Vector is only 10 meters because I subtracted 10 from my initial twenty.
The same thing happens in the Y direction. If I start out here at point A and then travels
negative 20 in the Y direction, meaning 20 meters South, I start with negative 20 and
then if I choose to walk 20 meters up, that's positive 20, positive being the UP direction,
and my total displacement is Zero (0) because I ultimately wound up where I started and
we can prove this Mathematically because negative twenty plus twenty is equal to Zero (-20 +
20 = 0). From these examples you can see that if we
have Vectors from the same direction we add them, if they're pointing in opposite direction,
you wind up subtracting. So what happens when you start at point A? Walk 10 meters to the
right, stop at point B and then walk 10 meters North to stop at point C. What is my total
displacement? I don't subtract because I'm not walking
backwards, I don't add them because I'm not walking in the same direction. Think of
the ultimate displacement on this city grid as the expression as the crow flies. If a
crow started out at point A and flew over all the buildings to wind up at point C, the
crow is travelling at a diagonal and the total displacement is the exact distance between
A to C regardless of the path traveled. And this is where the idea of Vector Components
comes in. if you want to find the Total Distance traveled, you have to take you Vector 1 and
turn it into V1x and V1y, then you take your Vector 2 and turn it into V2x and V2y. And
then to get the total, we're going to use the Pythagorean Theorem.
First we have to find the total X so we'll take the sum of x is equal to X one plus X2
(ex=X1+X2). The sum of Y is equal to Y one plus Y two (eY=Y1+Y2). This will give us our
new X and our new Y. And then we'll use the Pythagorean Theorem,
which is the formula is x squared plus y squared is equal to z squared (X^2 + Y^2 = Z^2). You
may also see this as A squared plus B squared is equal to C squared (A^2 + B^2 = C^2). Since
we're solving for C or Z, we take the square root to cancel out the squared and that means
the ultimate resultant is simply the square root of A squared plus B squared (v(a^2+b^2
) ) or the square root of X squared plus Y squared (v(x^2+y^2 ) ).
Going back to our example, this is relatively simple because the first Vector V1x is equals
to 10 meters (V1x=10m), V1y is equal to Zero (V1y=0) because we've only traveled in the
x direction. The second Vector we have V2x is equal to zero (V2x=0) because we traveled
North so we have nothing in the X direction. The V2y is equals to 10 meters (V2y=10m) because
we traveled 10 meters North. Now be very careful, I'm using V to represent
Vector not Velocity. This is just our walking Vector. Adding our Vectors, X1 equals ten,
X2 is Zero so X is really just 10 meters. Y1 equals to zero, Y2 is ten, once again we
have ten meters. Plugging our numbers into this equation we
have the Resultant is equal to the square root of X squared or ten squared plus Y squared
which is ten squared again (R=v(?10?^2+?10?^2 )). Ten squared is a hundred plus a hundred
is equal to the square root of two hundred v200 . Remember on the MCAT you cannot use
a calculator. So we want to break this up into something that we can recognize.
I don't know the square root of two hundred but the square root of two hundred is equal
to the square root of two times the square root of a hundred. The square root of two
something you should memorize and that is one point four (1.4), the square root of a
hundred is equal to ten (10) and so our answer is one point four times 10 (1.4 x 10). We�ll
use the times ten trick moving our decimal over one space to the right and that gives
an the answer of fourteen meters total as the crow flies or Resultant of fourteen meters
(14m). If you're not comfortable with the Math
trick I used here, go back to my MCAT Math videos on my website at http://leah4sci.com/MCATMATH.
You can also download my MCAT Math Cheat Sheet a lot of the different Mathematical Values
that you have to memorize for the MCAT. Be sure to join me in the next video where
I take you through the process of adding vectors that are not clean in the x and y direction
making you use Trigonometry to calculate the X and Y components of each Vector before you
add them. Are you stuck on a specific MCAT topic? I
offer Private Online Tutoring where I focus on your needs to strengthen your individual
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