Given that the modulus of vector 𝐀
is 35 and the modulus of vector 𝐁 is 23 and the dot product between 𝐀 and 𝐁 is
equal to negative 805 root two divided by two, determine the measure of the smaller
angle between the two vectors.
In this question, we’re given some
information about vectors 𝐀 and 𝐁. And we’re asked to determine the
smaller angle between these two vectors. Sometimes in these questions we
like to sketch a picture of what’s happening. However, the information we’re
given about our vectors won’t allow us to sketch a picture. We don’t know the components of
vectors 𝐀 and 𝐁. Instead, we only know their modulus
and their dot product.
So we’re going to need to rely
entirely on our formula. Remember, this tells us if 𝜃 is
the angle between two vectors 𝐀 and 𝐁, then the cos of 𝜃 will be equal to the dot
product between 𝐀 and 𝐁 divided by the modulus of 𝐀 times the modulus of 𝐁. And we would find the value of 𝜃
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 doesn’t really matter
how we draw our vectors 𝐀 and 𝐁. If the value of 𝜃 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 we’ll see, that’s not
what’s happening in this question.
Let’s now find the smaller angle
between our two vectors 𝐀 and 𝐁. It solves the equation the cos of
𝜃 will be equal to the dot product between 𝐀 and 𝐁 divided by the modulus of 𝐀
times the modulus of 𝐁. In the question, we’re told the dot
product between 𝐀 and 𝐁 is equal to negative 805 root two over two, the modulus of
𝐀 is equal to 35, and the modulus of 𝐁 is equal to 23. So we can substitute these values
directly into our formula, giving us the cos of 𝜃 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 𝜃 is
negative 805 root two divided by two times 35 times 23. And if we were to evaluate 35 times
23, we would see it’s exactly equal to 805. So we can cancel these, leaving us
with the cos of 𝜃 is equal to negative root two over two.
And finally, we can solve for our
value of 𝜃 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 𝜃 is the inverse cos
of negative root two over two, which we can calculate is 135 degrees.