Video Transcript
Given that the measure of the
smaller angle between 𝐀 and 𝐁 is 150 degrees and the magnitude of 𝐁 equals 54,
determine the component of vector 𝐁 along 𝐀.
Okay, in this exercise, we have
these two vectors 𝐀 and 𝐁. And just to help us see how they
relate to one another, let’s assume that they lie in the 𝑥𝑦-plane. We might then draw these two
vectors like this. And we’re told that the smaller
angle between these two vectors, that’s this one here, we can call it 𝜃, is 150
degrees. We’re asked to determine the
component of vector 𝐁 along vector 𝐀.
Looking at our sketch though, we
might wonder if the answer isn’t zero, because it looks as though none of vector 𝐁
lies along vector 𝐀. Here we have to be careful though
because this phrase “along 𝐀” really means along the line that passes through
vector 𝐀, in other words this dashed line, where the line is assumed to have a
direction equal to the direction of vector 𝐀. In answering this question then,
we’re going to be calculating this distance here. That’s the component of vector 𝐁
along 𝐀.
To get started solving for this,
let’s recall that the scalar projection of one vector onto another is equal to the
dot product of those vectors divided by the magnitude of the vector being projected
onto. And this is also equal to the
magnitude of the first vector, here we’ve called it 𝐕 one, multiplied by the cosine
of the angle between the two vectors. It’s this form of the scalar
projection equation that we can make use of in this particular exercise. After all, we know the magnitude of
what we could call our first vector, the magnitude of 𝐁, and we also know the angle
between our vectors.
What we want to calculate then is
the magnitude of 𝐁 times the cos of 𝜃, or substituting in our given values 54
times the cos of 150 degrees. The cosine of this angle equals
exactly negative the square root of three over two. So our scalar projection equals 54
times negative root three over two, or in simplified form negative 27 root
three. This is the component of vector 𝐁
along vector 𝐀. And we can see from this result
that, in general, a scalar projection can be negative. In this case, that came from the
fact that the part of 𝐁 lying along the line through vector 𝐀 points in the
opposite direction as vector 𝐀.
