Consider the vector negative seven 𝐢 minus five 𝐣. Calculate the direction of the vector, giving your solution as an angle to the nearest degree, measured counterclockwise from the positive 𝑥-axis.
Okay, so let’s sketch this scenario. Here are our 𝑥- and 𝑦-axes, which define our coordinate plane. And the event we’ll be considering is negative seven 𝐢 minus five 𝐣. So let’s draw this in standard position.
Recall that 𝐢 is the unit vector in the 𝑥-direction and 𝐣 is the unit vector in the 𝑦-direction. Negative seven 𝐢 is then the vector pointing in the opposite direction to the vector 𝐢 and so opposite to the 𝑥-axis with length seven. And from this vector, we subtract five 𝐣 or add negative five 𝐣, where negative five 𝐣 is the vector opposite in direction to the vector 𝐣, so pointing in the opposite direction to the 𝑦-axis, whose magnitude is five, to get the vector negative seven 𝐢 minus five 𝐣.
So this is the vector we’re considering, what we asked about it. Well, we want its direction. And we want that direction as the measure of the angle measured counterclockwise from the positive 𝑥-axis. So here’s the positive 𝑥-axis. And we want the measure of the angle measured counterclockwise from the 𝑥-axis to our vector, so this angle here.
The first thing that we can note is that the direction is greater than 180 degrees. In fact, if we add another angle to our diagram, we can see that the direction we’re looking for is 180 degrees plus the measure of this new angle, which we’ve called 𝜃. And we can also see that it’s relatively straightforward to find the measure of angle 𝜃 because we have a nice right triangle already drawn.
Tan 𝜃 is the ratio of the length of the side opposite the angle 𝜃 to the length of the side adjacent to the angle 𝜃. The length of the side opposite the angle 𝜃 is the magnitude of negative five 𝐣, which is five. And the length of the side adjacent to the angle 𝜃 is the magnitude of the vector negative seven 𝐢, which is seven. The measure of angle 𝜃 is then the arc tan of this ratio, five over seven.
We can enter this expression into our calculators, making sure that we’re in degree mode, to get an answer of 35.5376 dot dot dot degrees. And so our direction is 180 degrees more than this. So we perform the addition to get 215.5376 dot dot dot degrees.
And we are required to give our solution to the nearest degree. So we have to round this direction angle up, in this case to 216 degrees. This is to the nearest degree, the direction of the vector negative seven 𝐢 minus five 𝐣 measured counterclockwise from the positive 𝑥-axis.