### Video Transcript

Angle between Two Vectors in
Space

In this video, weโre going to learn
how we can find the angle between any two vectors in space by using the dot
product. And weโll see how to do this in a
few situations, for example, given the component forms of a vector or given a
graphical representation of the vectors.

To do this, weโre first going to
need to recall a couple of facts about vectors. First, we know how to find the dot
product of two vectors of equal dimensions. If ๐ฎ is the vector with components
๐ฎ one, ๐ฎ two, up to ๐ฎ ๐ and ๐ฏ is the vector with components ๐ฏ one, ๐ฏ two, up
to ๐ฏ ๐, then the dot product between ๐ฎ and ๐ฏ is equal to the sum of the products
of the corresponding components. ๐ฎ dot ๐ฏ is ๐ฎ one ๐ฏ one plus ๐ฎ
two ๐ฏ two and we sum all the way up to ๐ฎ ๐ times ๐ฏ ๐. And weโve seen a few different ways
we can apply the dot product.

For example, if ๐ is the angle
between vectors ๐ฎ and ๐ฏ, then we know the cos of ๐ will be equal to the dot
product between ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of
๐ฏ. And itโs worth pointing out thereโs
a second way of viewing this formula. If we let ๐ฎ hat be the unit vector
pointing in the same direction as vector ๐ฎ and ๐ฏ hat be the unit vector pointing
in the same direction as vector ๐ฏ โ so ๐ฎ hat is ๐ฎ divided by the magnitude of ๐ฎ
and ๐ฏ hat is ๐ฏ divided by the magnitude of ๐ฏ โ then the cos of ๐ will also be
equal to the dot product between ๐ฎ hat and ๐ฏ hat. This gives us a nice geometric
interpretation of the dot product.

This formula is what weโre going to
use to find the angle between our two vectors. Weโll calculate this expression and
then take the inverse cos of both sides of the equation. However, there is one thing worth
pointing out here about our value of ๐. We recall if weโre working in
degrees, then the inverse cosine function will have a range between zero and 180
degrees inclusive. So if we only take the inverse cos
of this expression, our answer will always be between zero and 180 degrees
inclusive.

And this has a useful result
geometrically. If we sketch the vectors ๐ฎ and ๐ฏ
starting at the same point, then by using this formula to find the value of ๐, we
will always get the smaller angle. And of course, we can find the
other angle directly from the sketch. These two angles sum to give us
360. So its angle will be 360 degrees
minus ๐. And an alternative method of seeing
why this is true is to think about what happens when we take the inverse cos of both
sides of the equation.

We know thereโs multiple solutions
for this. And we also know that if ๐ is a
solution to this, then 360 minus ๐ is also a solution because the cos of ๐ is
equal to the cos of 360 minus ๐. The last thing weโre going to point
out is that all of what we have just discussed is true if instead we were working in
radians. However, our values of ๐ would
range between zero and ๐ instead. Letโs now see some examples of how
weโre going to apply this to find the angle between two vectors.

Given that the modulus of vector ๐
is 35 and the modulus of vector ๐ is 23 and the dot product between ๐ and ๐ is
equal to negative 805 root two divided by two, determine the measure of the smaller
angle between the two vectors.

In this question, weโre given some
information about vectors ๐ and ๐. And weโre asked to determine the
smaller angle between these two vectors. Sometimes in these questions we
like to sketch a picture of whatโs happening. However, the information weโre
given about our vectors wonโt allow us to sketch a picture. We donโt know the components of
vectors ๐ and ๐. Instead, we only know their modulus
and their dot product.

So weโre going to need to rely
entirely on our formula. Remember, this tells us if ๐ is
the angle between two vectors ๐ and ๐, then the cos of ๐ will be equal to the dot
product between ๐ and ๐ divided by the modulus of ๐ times the modulus of ๐. And we would find the value of ๐
by taking the inverse cosine of both sides of this equation. And this gives us a useful result
because the inverse cosine function has a range between zero and 180 degrees.

Therefore, it doesnโt really matter
how we draw our vectors ๐ and ๐. If the value of ๐ is between zero
and 180, it will always give us the smaller angle between these two vectors. The only possible caveat to this
would be if of our vectors point in exactly opposite directions. Then the angle measured in both
directions will be equal to 180 degrees. However, as weโll see, thatโs not
whatโs happening in this question.

Letโs now find the smaller angle
between our two vectors ๐ and ๐. It solves the equation the cos of
๐ will be equal to the dot product between ๐ and ๐ divided by the modulus of ๐
times the modulus of ๐. In the question, weโre told the dot
product between ๐ and ๐ is equal to negative 805 root two over two, the modulus of
๐ is equal to 35, and the modulus of ๐ is equal to 23. So we can substitute these values
directly into our formula, giving us the cos of ๐ is negative 805 root two over two
all divided by 35 times 23.

We can simplify this. Remember, dividing by a number is
the same as multiplying by the reciprocal of that number, giving us the cos of ๐ is
negative 805 root two divided by two times 35 times 23. And if we were to evaluate 35 times
23, we would see itโs exactly equal to 805. So we can cancel these, leaving us
with the cos of ๐ is equal to negative root two over two.

And finally, we can solve for our
value of ๐ by taking the inverse cos of both sides of the equation. Remember, we know this will give us
the smaller angle between our two vectors. This gives us ๐ is the inverse cos
of negative root two over two, which we can calculate is 135 degrees.

Letโs now see an example of how we
would calculate the angle between two vectors given their component forms.

Find the angle ๐ between the
vectors ๐ four, two, negative one and ๐ eight, four, negative two.

In this question, weโre given two
vectors in component form. And weโre asked to find the angle
๐ between these two vectors. To do this, we know a formula for
finding the angle between any two vectors. We recall if ๐ is the angle
between vectors ๐ and ๐, then ๐ satisfies the equation the cos of ๐ is equal to
the dot product between ๐ and ๐ divided by the modulus of ๐ times the modulus of
๐.

And since weโre given ๐ and ๐ in
component form, we can calculate all of these values. So we can find our value of ๐. Letโs start by calculating the dot
product between ๐ and ๐. So we want to find the dot product
between the vectors four, two, negative one and eight, four, negative two. Remember, to find a dot product
between two vectors, we need to multiply the corresponding components together and
then add all of these together.

Multiplying the first components of
each vector together, we get four times eight. Multiplying the second components,
we get two times four. And multiplying the third
components, we get negative one times negative two. So the dot product of these two
vectors will be the sum of these three products. And we can evaluate this
expression. We get the dot product between ๐
and ๐ is 42.

However, this is not the only thing
we need to calculate. We also need to find the modulus of
๐ and the modulus of ๐. To do this, weโre first going to
need to recall how we find the modulus of a vector. Remember, the modulus of a vector
is the square root of the sum of the squares of its components. In other words, the modulus of the
vector ๐, ๐, ๐ is the square root of ๐ squared plus ๐ squared plus ๐
squared. And we know the components of ๐
are four, two, and negative one. So the modulus of ๐ is the square
root of four squared plus two squared plus negative one squared.

And if we evaluate this expression,
we see itโs equal to the square root of 21. We can then do exactly the same
thing to find the modulus of ๐. Itโs equal to the square root of
eight squared plus four squared plus negative two all squared. And if we were to evaluate and
simplify this expression, we would see that the modulus of ๐ is two root 21. Now that weโve found the dot
product between ๐ and ๐ and the modulus of ๐ and the modulus of ๐, we can
substitute these into our equation involving ๐.

We showed the dot product between
๐ and ๐ is 42, the modulus of ๐ is root 21, and the modulus of ๐ is two root
21. Therefore, the cos of ๐ is 42 over
root 21 times two root 21. However, if we start evaluating
this expression, we see something interesting. In the denominator of this
expression, root 21 multiplied by two root 21 simplifies to give us 42. And 42 over 42 is one. So our entire equation simplifies
to give us the cos of ๐ is equal to one, and we can solve for ๐. We take the inverse cosine of both
sides of the equation, giving us ๐ is the inverse cos of one, which we know is zero
degrees.

And we could stop here; however,
this gives us a useful piece of information. If the angle between ๐ and ๐ is
zero degrees, then ๐ and ๐ point in the same direction. In other words, weโve also shown
that the vectors ๐ and ๐ are parallel. And in fact, we could directly
prove this. We would see that our vector ๐ is
just two times the vector ๐. And thereโs a useful result we can
get from this. Because our scalar is positive, the
angle between these two vectors will be zero. However, if this scalar was
negative, then the angle between them would be 180 degrees because then our vectors
would point in exactly opposite directions. Either way, we were able to show
the angle ๐ between the vectors ๐ and ๐ given to us in the question was zero
degrees.

Letโs see another example of
finding the angle between two vectors.

Find the angle ๐ between the
vectors ๐ฏ is equal to ๐ข and ๐ฐ is equal to three ๐ข plus two ๐ฃ plus four ๐ค. Give your answer correct to two
decimal places.

In this question, weโre given two
vectors ๐ฏ and ๐ฐ in terms of the unit directional vectors ๐ข, ๐ฃ, and ๐ค. And weโre asked to find the angle
๐ between these two vectors. And we need to give our answer to
two decimal places.

To do this, we can start by
recalling we have a formula to find the angle between two vectors. Since ๐ is the angle between
vectors ๐ฏ and ๐ฐ, the cos of ๐ must be equal to the dot product between ๐ฏ and ๐ฐ
divided by the magnitude of ๐ฏ times the magnitude of ๐ฐ. So to find the value of ๐, we need
to find the dot product between ๐ฏ and ๐ฐ, the magnitude of ๐ฏ, and the magnitude of
๐ฐ. Then all weโll need to do is take
the inverse cosine of both sides of the equation.

Thereโs several different ways of
doing this. For example, we could work directly
with the unit directional vector notation for ๐ฏ and ๐ฐ. However, we could also write these
vectors component-wise by taking the coefficients of the unit directional
vectors. Either method will work; itโs
personal preference which weโll use.

Weโll write ๐ฏ and ๐ฐ
component-wise. ๐ฏ is the vector one, zero, zero
and ๐ฐ is the vector three, two, four. Letโs now start finding the values
of our equation. Letโs start with the dot product
between ๐ฏ and ๐ฐ. Remember, to find the dot product
of two vectors, we need to find the product of the corresponding components and then
add the results together. In this case, the first component
of ๐ฏ times the first component of ๐ฐ is one times three, the second component of ๐ฏ
times the second component of ๐ฐ is zero times two, and the third component of ๐ฏ
times the third component of ๐ฐ is zero times four.

So the dot product is the sum of
these, one times two plus zero times two plus zero times four. And we can just calculate this
expression. The second and third terms are
zero. So this just gives us three. We now want to find the magnitude
of our two vectors. Letโs start with the magnitude of
๐ฏ. Remember, ๐ฏ is the unit
directional vector ๐ข. And remember, ๐ข is a unit
directional vector. It has magnitude one.

Now to find the magnitude of ๐ฐ is
more difficult. So weโll write this in component
notation. And weโll recall to find the
magnitude of a vector, we need to find the square root of the sum of the squares of
the components. So the magnitude of the vector ๐,
๐, ๐ will be the square root of ๐ squared plus ๐ squared plus ๐ squared. So for vector ๐ฐ, this will be the
square root of three squared plus two squared plus four squared, which we can
calculate is equal to the square root of 29.

Now that we found these values, we
can substitute them into our equation for ๐. We showed the dot product between
๐ฏ and ๐ฐ is three, the magnitude of ๐ฏ is one, and the magnitude of ๐ฐ is root
29. So we must have the cos of ๐ is
three divided by one times the square root of 29. And now we can solve the ๐ by
taking the inverse cos of both sides of our equation. This gives us that ๐ is the
inverse cos of three divided by root 29. And if we calculate this and round
our answer to two decimal places, we see that ๐ is 56.15 degrees.

Letโs now see an example of how we
would find the angle between two vectors given in a diagram.

Find the angle between the vectors
shown in the following diagram.

In this question, we need to find
the angle between two vectors. And weโre given these vectors on a
diagram. And we can also see the angle
between the two vectors given on our diagram. We have a few different options of
how we could calculate this. For example, we could just do this
by using trigonometry. However, we also have a formula for
finding the angle between two vectors. We recall if ๐ is the angle
between two vectors ๐ฎ and ๐ฏ, then the cos of ๐ will be equal to the dot product
of ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of ๐ฏ. We can then use this to solve for
the value of ๐ by taking the inverse cosine of both sides of the equation.

So to answer this question, weโre
going to need to find the dot product between our vectors ๐ฎ and ๐ฏ and the
magnitude of ๐ฎ and the magnitude of ๐ฏ. And to do this, weโre going to want
to write our vectors in component form. Weโll do this by using the
diagram. Letโs start with our vector ๐ฎ. We could see on our diagram it
starts at the origin and then at the endpoint it has an ๐ฅ-coordinate of negative
two root three. So the change in ๐ฅ is negative two
root three. Similarly, we can see its
๐ฆ-coordinate starts at zero and ends at two and its change in ๐ฆ is two. So we can represent ๐ฎ as the
vector with horizontal component negative two root three and vertical component
two.

We can do exactly the same for
vector ๐ฏ. We can see it starts at the origin
and then ends at an ๐ฅ-coordinate of negative two and it starts at the origin and
ends at a ๐ฆ-coordinate of negative two. So the change in ๐ฅ is negative
two, and the change in y is negative two. So ๐ฏ is the vector negative two,
negative two. Now we need to find the dot product
of these two vectors and their magnitudes.

Letโs start by calculating the dot
product between ๐ฎ and ๐ฏ. Remember, to find the dot product
of two vectors, we need to find the products of corresponding components and then
add all of these together. So we multiply the first components
of ๐ฎ and ๐ฏ together to give us negative two root three times negative two. And then we add the products of
their second components. Thatโs two multiplied by negative
two. And if we calculate this
expression, we see itโs equal to four root three minus four.

But weโre not done yet. We still need to find the magnitude
of ๐ฎ and the magnitude of ๐ฏ. Letโs start by finding the
magnitude of ๐ฎ. Remember, we can find this by
taking the square root of the sums of the squares of the components. So the magnitude of ๐ฎ is the
square root of negative two root three all squared plus two squared, which, we can
calculate, gives us the square root of 12 plus four, which is root 16, which is of
course just equal to four. We can then do exactly the same to
find the magnitude of ๐ฏ. We square each component of ๐ฏ, add
these together, and take the square root. The magnitude of ๐ฏ is the square
root of negative two squared plus negative two squared, which of course simplifies
to give us the square root of four plus four, which is equal to root eight.

And now that we found all of these
values, weโre ready to substitute them into our equation for ๐. Substituting in ๐ฎ dot ๐ฏ is four
root three minus four, the magnitude of ๐ฎ is four, and the magnitude of ๐ฏ is root
eight, we get the cos of ๐ is four root three minus four all divided by four root
eight. And itโs worth pointing out here we
can simplify this expression to give us root six minus root two all divided by
four. However, itโs not necessary because
all we need to do now is take the inverse cosine of both sides of the equation.

This gives us that ๐ is the
inverse cos of root six minus root two all divided by four, which, we can calculate,
gives us 75 degrees. And this is our final answer
because if we look in our diagram, there are two possible angles between vectors ๐ฏ
and ๐ฎ. Thereโs the angle shown in our
diagram and thereโs the angle we could take in the opposite direction. However, this secondary angle shown
in green is bigger than 75 degrees, so it canโt possibly be 75 degrees. In fact, it would be 360 minus 75
degrees. Therefore, we were able to show the
angle between the two vectors in our diagram ๐ฎ and ๐ฏ is given by 75 degrees.

Letโs now go through one last
example of how we can use our formula to find information about vectors.

The angle between vector ๐ and
vector ๐ is 22 degrees. If the magnitude of vector ๐ is
equal to three times the magnitude of vector ๐ is equal to 25.2, find the dot
product between ๐ and ๐ to the nearest hundredth.

In this question, weโre given some
information about two vectors ๐ and ๐. First, weโre told the angle between
these two vectors is equal to 22 degrees. Next, weโre also told information
about their magnitudes. We know the magnitude of ๐ is
equal to 25.2, and we know that three times the magnitude of ๐ is also equal to
25.2. So the magnitude of ๐ is three
times bigger than the magnitude of ๐. We need to use this to find the dot
product of ๐ and ๐. And we need to give our answer to
the nearest hundredth.

To answer this question, we need to
notice that we know a formula which connects the angle between two vectors with
their dot product. We recall if ๐ is the angle
between two vectors ๐ and ๐, then we know that the cos of ๐ must be equal to the
dot product between ๐ and ๐ divided by the magnitude of ๐ times the magnitude of
๐. And in this question, we already
know some of these values. For example, weโre told the angle
between our two vectors is 22 degrees. Next, weโre also told the magnitude
of ๐ is equal to 25.2.

And we could also find the
magnitude of ๐ using the information given to us in the question. One way of doing this is to notice
that three times the magnitude of ๐ is equal to 25.2. We can then solve this to find the
magnitude of ๐ by dividing both sides of our equation through by three. And calculating this, we get the
magnitude of ๐ is 8.4. So, in fact, the only unknown in
this equation is the dot product between ๐ and ๐. And thatโs exactly what weโre asked
to calculate.

So weโll substitute the angle of ๐
equal to 22 degrees, the magnitude of ๐ equal to 25.2, and the magnitude of ๐
equal to 8.4 into our equation. This gives us the cos of 22 degrees
should be equal to the dot product between ๐ and ๐ divided by 25.2 times 8.4. And now we can just rearrange this
equation for the dot product between ๐ and ๐. We multiply through by 25.2 times
8.4. This gives us ๐ dot ๐ is 25.2
times 8.4 times the cos of 22 degrees. And we can calculate this to the
nearest hundredth or to two decimal places. Itโs equal to 196.27.

Letโs now go over the key points of
this video. First, we know if ๐ is the angle
between two vectors ๐ฎ and ๐ฏ, then the cos of ๐ will be equal to the dot product
of ๐ฎ and ๐ฏ divided by the magnitude of ๐ฎ times the magnitude of ๐ฏ. And this works so long as neither
vector ๐ฎ or ๐ฏ is equal to zero. And to take the dot product of ๐ฎ
and ๐ฏ, we need them to have the same dimension. We can also use this formula to
find the angle between two vectors by taking the inverse cosine of both sides of
this equation. But we do need to be careful
because the inverse cosine function has a range between zero and 180 degrees
inclusive, or if weโre working in radians, zero to ๐ inclusive. So this method will give us the
smaller of the two angles between our vectors ๐ฎ and ๐ฏ.