Question Video: Finding the Direction Angle of a Given Vector Mathematics

Video Transcript

Consider the vector negative two, three. Calculate the direction of the vector, giving your solution as an angle to the nearest degree measured counterclockwise from the positive 𝑥-axis.

Since we’re given the components of our vector, let’s start out by plotting it on this 𝑥𝑦-plane. We’ll say that each of these tick marks represents a distance of one unit. And we’re told that our vector has an 𝑥-component of negative two and a 𝑦-component of positive three. That tells us that with its tail at the origin our vector will look like this. We want to calculate the direction of this vector, and we’ll do it in terms of an angle that begins at the positive 𝑥-axis and goes to our vector. We can give this angle a name. We’ll call it 𝜃. At this point, let’s recall that in general when a vector, we can call it 𝐕, is given in polar form, then that vector is defined by a radial distance from the origin and an angle from the positive 𝑥-axis.

In our exercise, we want to solve for this angle 𝜃. And to do this, we’re given a vector not in polar form but in rectangular form. In using this 𝑥 and 𝑦 information then to solve for 𝜃, we’re effectively converting from a rectangular form of a vector to part of its polar form. We say all that because the tan of 𝜃, where 𝜃 is the angle of a vector in its polar form, is equal to the ratio of the rectangular components of that vector, 𝑦 to 𝑥. This tells us that the tan of our angle of interest, 𝜃, is equal to three divided by negative two. And then if we take the inverse or arctan of both sides, the left-hand side simplifies to the angle we want to solve for.

Now, if we go and evaluate this expression on our calculator to the nearest hundredth of a degree, we find a result of negative 56.31 degrees. Looking at our sketch though, we see that this can’t be the angle 𝜃. What’s happening here is because the 𝑥-value in this fraction is negative — in other words, the vector we’re working with occupies either the second or the third quadrant — to correctly compute our angle, we’ll need to take our calculated result and add 180 degrees to it. For any vector with a negative 𝑥-component like we have here, this is a standard practice for accurately calculating its direction.

And so now if we add these two angles together, we get a result of 123.69 degrees. And recalling that we give our answer rounded to the nearest degree, we can see that 𝜃 equals 124 degrees. This is the direction of our vector to the nearest degree measured counterclockwise from the positive 𝑥-axis.