Find the magnitude ||v||, to the nearest hundredth, and the di...

Question

Find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v.

v = 2i - 8j

Accepted Answer

Welcome back everyone. The magnitude and direction angle of the vector A given below express your answer to one decimal place. The vector is given to A equals 12, I minus 18 J. And there are four answer choices as 21.6 and negative 56.3 degrees. B says 31.6 and 56.3 degrees C 21.6 and 56.3 degrees and D 31.6 and negative 56.3 degrees. So we're given our vector and essentially it has a component form such that A equals A X, we have our X component I minus A Y component J, right. So whenever we want to identify the magnitude, we can simply use the magnitude formula, we can essentially state that the magnitude of a vector A is equal to square root of A X squared plus A Y squared. We're taking those individual components and we can see that A X is equal to 12. That's the number in front of I and A Y is negative 18, right? We have to take that negative sign into account. So we can immediately identify the magnitude we're going to take square root of 12 squared plus negative 18 squared. All that we have to do is simply evaluate the result, we're going to take square root of 400 in 68. And now we want to express the answer to one decimal place. So that's approximately equal to 21.6. Now, considering the direction it is really important to understand where this vector is approximately, right. So if we visualize the coordinate plane, we can see that the A X component is positive and the A Y component is negative. So this means that we have the fourth quadrant. And whenever we have 1/4 quadrant, we can essentially estimate that angle. Let's call it alpha as the following expression, alpha is equal to the inverse of 10 of A Y divided by A X. We can apply this formula for the first and the fourth quadrant. So in this case, we simply want to take the inverse of 10 of negative 18 divided by 12. Now, if we evaluate the result and if we round it to one decimal place, we're going to get negative 56.3 degrees. And now we have our direction, we also have the magnitude and we can see that the correct answer to this problem corresponds to the answer choice. A thank you for watching.

Find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v.

v = 2i - 8j

Key Concepts

Magnitude of a Vector

Direction Angle of a Vector

Quadrants in the Cartesian Plane

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