Consider the vector 𝐕 equal to three, negative two. Which of the following graphs accurately represents the vector? Calculate the modulus of the vector. Given that positive numbers represent measuring counterclockwise, calculate the measure of the angle the vector makes with the positive 𝑥-axis. Give your answer to one decimal place between negative 180 degrees and 180 degrees.
We begin by considering the vector given in the question three, negative two. This is given in the rectangular form 𝑥, 𝑦, where the values three and negative two represent the displacement in the 𝑥- and 𝑦-directions, respectively. From the origin, we travel three units in the positive 𝑥-direction, which rules out options (A), (C), and (D). The 𝑦-component of our vector is negative two. So we travel two units in the negative 𝑦-direction. This takes us to the point with coordinates three, negative two. We can therefore conclude that the graph that represents the vector three negative two is graph (E).
We will now clear some space in order to answer the second and third parts to the question. The second part of the question asks us to calculate the modulus of the vector. We recall that the modulus or magnitude of a vector is its length. This is denoted as shown. And the magnitude is equal to the square root of 𝑥 squared plus 𝑦 squared, where 𝑥 and 𝑦 are the components of the vector. Substituting in our values of 𝑥 and 𝑦, we have the square root of three squared plus negative two squared. Since three squared is equal to nine and negative two squared is equal to four, then the magnitude or modulus of the vector is equal to root 13. We could also have calculated this by creating a right triangle on our graph and using our knowledge of the Pythagorean theorem. Once again, we would have got an answer of root 13.
The final part of this question asks us to calculate the measure of the angle that the vector makes with the positive 𝑥-axis. We are told that positive numbers represent measuring counterclockwise. And since this angle is measured in a clockwise direction from the positive 𝑥-axis, our answer will be negative. We can also see from our graph that this will lie between zero and negative 90 degrees. If we begin by considering the angle 𝛼 in our right triangle, we can calculate this using our knowledge of right angle trigonometry. We know that the tangent of any angle 𝜃 in a right triangle is equal to the opposite over the adjacent. This means that in our triangle, tan 𝛼 is equal to two-thirds.
Taking the inverse tangent of both sides, we have 𝛼 is equal to the inverse tan of two-thirds. Ensuring that our calculator is in degree mode, typing in the right-hand side gives us 𝛼 is equal to 33.69 and so on degrees. Rounding this to one decimal place gives us 33.7. And as already mentioned as this angle must be negative, the measure of the angle the vector makes with the positive 𝑥-axis is negative 33.7 degrees.
We now have answers to all three parts of this question. They are graph (E), root 13, and negative 33.7 degrees.