Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. (d) a = (20i + 10j) m/s^2

Video transcript

Hey, everyone in this problem, we're asked to represent the vector A which is equal to 10 I plus six J graphically and then to indicate its direction and magnitude in the image. So looking at the diagrams that we have, we have four answer choices A through D, they actually all show the same vector drone. And that vector starts at the origin 00 and ends at the 0.10 comma six. And that makes sense, we have the vector 10 I plus six J. So the X component is 10, the Y component is six. And that's exactly what's drawn in all of these diagrams. Now what differs in these diagrams is the angle and the magnitude. So let's go ahead and calculate those two things. And let's start with the angle. Now, we know if we were to draw a triangle out of our, this where vector A will be the hypo. So Y component is gonna be the opposite side. The component is gonna be the adjacent side, we can relate the opposite and adjacent sides through the tangent of theta. So we can write that tangent of beta is equal to the opposite side divided by the adjacent side. I recall that trigonometric ratio. No, we can write that 10 data is equal to the opposite side. Again, that's the Y component which is six, the adjacent side, the X component is 10. So 10 data is equal to six divided by 10. So we take the inverse tangent, we get that theta is equal to 30.96 degrees to be round to the nearest degree like the diagrams do we have the beta is about degrees. Now, option B has that the angle is 59 degrees. So that's not correct. So we can eliminate option B and now we're looking at either AC or D and what it's gonna come down to is the magnitude. Now we call it the magnitude of a vector is we write these double lines with a in between gonna be the square root of the X component of A squared. What's the Y component of a work? We have the square root of A X squared plus A Y squared. Working this though to get the square root of the X component is 10, so 10 squared plus the Y component, six square. So simplify, we have 100 plus 36 under our square root, we have the square root of 136. If we use our calculator to estimate this value, we get that A, the magnitude of A is approximately 11.662. So once again, looking at our answer traces, we can now see that the correct answer is option D it has the vector drawn correctly pointing from the origin to the point at 10 comma six. It has the angle of 31 degrees and the magnitude of 11.7. Thanks everyone for watching. I hope this video helped see you in the next one.