in this video we discuss scalar multiples of vectors how to express a horizontal or vertical
vector as a scalar multiple of a unit vector and how to express any vector as a linear combination
of unit vectors we also discussed several notations for the components of a vector and
then we finish with a couple simple examples so we begin by discussing scalar multiplication and
a scalar is just a number and when we multiply a vector by a scalar it simply scales that vector or
stretches it so in our first example we're given a vector v and we're asked to draw the vector 2v and
one way to answer this correctly is to say that you just stretch it by a factor of 2 and leave the
direction the same and there's 2v but i could also go back to fundamentals here and say what does it
mean to multiply something by 2 and what it means to say 2v is add v to itself twice that's v plus
v so that's just another way to look at it and if i vector add v plus v i put the tail of the second
on the head of the first i draw a vector from the tail with the first to the head of the second and
i get the exact same answer of a vector twice as long and pointing in the original direction
next we take a look at some unit vectors so a unit vector in general is just any vector that has a
length of one but we're particularly interested in the unit vectors that point along the coordinate
axes so first is our horizontal unit vector again that has a length of one and it points in the
positive x direction and that gets a special name is an i hat the next one's a vertical unit vector
that has a length of one and points in the positive y direction and we call that j hat and
if we're in a three dimensional space we would just have another unit vector pointing in the
z direction and that's called k hat in the next example we start to get a sense for why these unit
vectors might be useful we're asked to express the following vectors as multiples of i hat or j
hat and the first one is a vector of length five pointing exactly in the positive x direction well
that's just a scalar multiple of the unit vector that's pointing in the positive x direction it's
just five times i hat the next one points in the positive y direction and it has a length of three
well that's just j hat stretched by a factor of three so that's three j hat and we can do
this for any vector it doesn't have to be pointing perfectly horizontal or vertical
in the next example we have a tilted vector and i've labeled the magnitudes of the components here
so the x component is a vector of length 10 pointing in the positive x direction and the
y component is a vector of length 6 pointing in the plus y direction each of those vectors can be
represented as a scalar multiple of a unit vector and so i could write v as 10 i hat plus 6 j hat
so this is true for any two-dimensional vector i could write it as a linear combination of the
unit vectors i hat and j hat and that phrase linear combination just means a sum of constant
multiples of those things next we're going to quickly summarize some different notations for
two-dimensional vectors and all of these notations extend naturally to the three-dimensional case
you would just tack on a z component and the unit vector pointing in the z direction is called k hat
so here's a few different ways of writing the vector v we could start with the unit vector
notation you write it as a linear combination of i hat and j hat alternatively i can just make a
list of numbers telling me the x and y coordinates of that vector and if it was a 3d vector i would
tell you the x y and z coordinates so in the second notation i've done that as a row vector
so it's a list of numbers horizontally separated by commas wrapped in parentheses now there are
other notations for this it's perfectly fine if you want to use brackets and skip the commas
and a lot of math textbooks use pointy brackets as well now the argument for using pointy brackets is
to distinguish a vector from an ordered pair but in physics we typically just use this parentheses
notation and we don't get confused with ordered pairs because we understand by context
that we're talking about an arrow a vector and not just a point in space
finally we have the column vector notation so you can just list the coordinates of your vector
vertically and wrap it in brackets or parentheses or in some math textbooks giant pointy brackets
in our example we're asked to write this vector v in terms of unit vectors and then
as a row vector and then as a column vector so this v is going to be 2 i hat plus 3 j hat
or as a row vector it's just a list of numbers 2 comma 3 or as a column vector i prefer to use
brackets for those it's just the number 2 and then the number 3 on the bottom and wrap it in brackets
so let's wrap things up with a couple extra examples in the first one i'm asked to express
this vector as a linear combination of unit vectors and this time i'm not given the components
i have to compute those myself so we have to do a little bit more work and that x component
is given by the hypotenuse times the cosine of the angle into three sig figs that gives me 96.4
and the y component will be hypotenuse times the sine of the angle into three sig figs that gives
me 115 and i was asked to write this vector as a linear combination of unit vectors so i have
96.4 i hat again that's a scalar multiple of the unit vector pointing in the positive x direction
that gives me a length of 96.4 pointing to the right and then i vector add to that 115
in the j hat direction finally we're asked to compute the magnitude and
direction of a vector that's given as a column vector 1 3. and it would be
nice to have a picture of this thing so my vector has components of length 1 and 3.
one pointing to the right three pointing up and i'm interested in the magnitude of this guy
and the angle with respect to the x-axis and so the magnitude is just given by the pythagorean
theorem i take the square root of the sum of the squares of the two legs of that right triangle
and i get square root of 10 out of this and the square root of 10 is about 3.16
then we want the angle and that's given by an inverse tangent the angle whose tangent
is three over one into three sig figs that's 71.6 degrees if you enjoyed this video or at
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