BAM! Mr. Tarrou. In this problem we are going to combine our knowledge of
geometrically representing vectors finding the resultant or the sum of
vectors using a triangular method and doing some basic algebraic operations
with vectors or simply just adding and subtracting them and we're going to do
this we're going to be given some vertices of this cuboid it may also be a
parallelepiped in your problem and from those vertices we're gonna find vector
AC ad and CG and then we have part two of this question that says hence
determine the position vectors of B EF and vertice H well hence means that for
Part B you need to you have to if you want full credit use the information
that you were either given or that you found more than likely in a previous
part of the questions so to answer Part B we have C week we're going to have to
use the vectors that we found in Part A okay now you don't have to draw a
three-dimensional grat sort of graph or system if you will and place this
exactly where it belongs in space though if it helps you you can go ahead and do
that you can just use this information given and build off of that to answer
Part B and let me show you how we're first going to find vector AC okay well
that means that we're going to be doing the terminal each of the components of
the terminal point - each of the corresponding components of the initial
point so vector AC is going to be the result of doing the position vector OC -
the position vector for the coordinate of a or you may just be with your
textbook just straight up subtracting each of these components and even though
I'm writing this notation that's exactly what I'm going to do AC is going to be
equal to 2 minus 6 subtracting D x-components now the Y components
negative 3 - negative two and the Z components 5-3 so
a vector a C is going to be equal to negative 4 2 negative 3 plus 2 of course
is negative 1 and 2 vector ad well that is going to be going to be the result
again terminal minus initial each of those corresponding components so we
have 7 minus 6
0 minus negative 2 and 6 minus 3 given us that ad vector ad is equal to 1 2 & 3
and then vector C G I meant to color code these yeah and vector C G okay so
not the same initial point of a now so pay attention to what you're doing
terminal minus initial or the position vector for vector position vector for
coordinate G and the minus the position vector for coordinate C so vector o G
minus vector both C or just subtracting the components 3 minus G C okay so 3
minus 2 trying to make a careless mistake
negative 5 that's negative 3 6 minus 5 so vector C G is equal to 1 negative 5
plus 3 is negative 2 & 1 and let me just make sure I haven't made
some kind of silly mistake No okay so now when I get to Part B and it says
hence use these three vectors to find the position vectors so we're going to
find vector OB vector Oh II Oh F and finally vector
OAH using these three vectors let's make sure that you know how to use that
triangular method process to add multiple vectors and move along space
throughout space and so to do this make sure I understand really what I'm doing
I'm going to go ahead and color code these vectors in our cuboid so we have
vector AC so we have vector a which is negative 4 negative 1 2 which is going
to take us in space from the front left corner of this cuboid down to the back
right lower corner and I'm going to count on the color coding here to keep
track of what I'm doing vector ad so if I want to go from
vertice a to vertice d I'm going to move to the forward one on the x-axis to the
right 2 on the y-axis and up 3 so this is vector ad I'm hoping that the green
differentiate H itself enough from the blue up here and then finally a vector C
G going from vector coordinate or vertices C to vertice ghee okay now how
can I use these three vectors the green the white and the orange ad AC and
vector C G to create the or write the position vector for coordinate beef so I
have to start at the origin now two of my given vertices start from vertex a
and I want to move through this cuboid or maybe a parallel paper on your
particular problem depends on what you know your book writes out so I'm gonna
just say hey let's get from the origin to a and then use these three vectors to
move around this cuboid to get whatever to get to whatever vertice we need so
we're going to do first vector Oh a okay so we move wherever
this is in space there's an origin and we move to vertice a and we want to get
to vertice B okay so if I'm here in my cuboid and I want to move to this vertex
or vertice of B only using these three
vectors well there's no given niat none of these three vectors simply move from
the left hand side to the right hand side and yeah I'd be nice to say well
it's vector o a plus vector a B but then I'm hence I'm not using one of these
three vectors that wouldn't get you full credit that's why I'm not just saying
it's Oh a plus a B so we're gonna go from somewhere the origin to a how about
we get back to C because that vector AC is going to take us from the left-hand
side to the right-hand side of this cuboid so we have plus a C so we're
somewhere in the origin move to a back to C we need to go towards back towards
B well we have a vector that goes from front to back so let's just subtract
that and so that would take us instead of that would be the opposite vector
instead of going from front to back it would move us from back to front no
because it says hence use these vectors exactly you know you could say minus ad
but I like this I'm kind of wanting to write these out so that these letters
attach you go from o to a the origin to a from A to C and then I want some
notation that takes me from vertex C to vertex B so since this is a cuboid and
thus is kind of telling us that these what appear to be parallel sides
actually are going to be parallel we're going to say that this is that C B is
equal to or that let's understanding that AED is equal to CB we
can say that this is Oh a vector A plus vector a C minus C D so from the origin
to a from A to C and now Ramone we've been back in the opposite direction
we're subtracting or using the opposite vector ad which is CB and that is going
to be equal to I'm definitely run out of room as I do
this example position vector for a 6 negative 2 3 x axis y axis z axis
movement vector AC well we just found that vector AC is negative 4 negative 1
2 and then finally subtract vector C D which is the same as vector ad well ad
is 1 2 3 so we're going to subtract vector C D which is equivalent to ad and
all this comes out to OB is equal to 6 plus negative 4 is 2 minus 1 is 1
negative 2 plus negative 1 is negative 3 now minus 2 is negative 5 and 3 plus 2
is 5 minus 3 is equal to 2 and one more time just for clearing let's bring this
notation over here this is the position vector for vector OB and I do want to
check my work just to make sure I'm not doing something silly ok
Oh e well Oh II let's see here just pick a different color to differentiate it
again let's get from the origin to a get a starting point there okay this is
going to take us from the origin to vertex a now we need to get to e well
we'd have to go up and we have a vector that moves us up vector CG is a vertical
component of this cuboid so we're going to do Oh a plus C G but as this is a
cuboid it'd be the same as if it was a parallelepiped CG is equivalent to AE
and that means that Oh II is equal to actually you know what it's not a great
idea really to write an equal sign next to every line of work let's do scratch
work and then finalize your answer when you're done Oh a plus AE which is
equivalent to CG and we found that already hence hence using that previous
information and therefore Oh II 6 plus 1 is 7 negative 2 plus negative 2 is
negative 4 and 3 plus 1 is equal to positive 4 and that is how you do that
last one so that's there's we're finding four coordinates four vertices it's
going to be similar to this so I'm going to step out
maybe you should pause the video try and find these other two the position
vectors for these other two vertices yourself and I'm going to pause the
video I'll just display the work for each of
these two answers quietly giving you a chance to try it out and that is the end
of our lesson
so we have position vector for Oh F is Oh a AC plus vector C G minus G F so
that meant again we went from something from the origin to a so now we're at a
then AC from C we went C G because we had to go from the bottom of the cuboid
to the top plane of that cuboid and then minus G F minus G F because we're trying
to go back to F now I went straight away and wrote minus vector G F but of course
we have vector ad is equivalent to vector F G ad is going from front to
back so we did the subtraction to go in the opposite direction and move from the
back of the keyboard to the front and then just filled in all the position
vectors and the vectors that we found in part a again the hence part of this
problem Oh H from origin to a from a to D and
then from finally D to H now mr true hey go to your homework
you