in this article we're going to talk about vectors so looking at this question which of the following quantities is not a vector would you say displacement velocity acceleration mass or force we need to be familiar with two things you need to be familiar with scalar quantities and vector quantities a scalar quantity is something that has magnitude but no direction for instance temperature is a scalar quantity it has a magnitude let's say if it's 80 degrees fahrenheit that's the magnitude of the temperature but you can't apply direction to temperature you can't say it's i mean you could say it but it doesn't make any sense if you were to say it's 80 degrees fahrenheit east that would be relevant force for instance is a vector because you can describe it using magnitude and direction for instance you can have a force of 100 newtons directed at an angle of 30 degrees above the x-axis so this is the magnitude of the force and this is the direction of the force which makes it a vector so anything that has both magnitude and direction is a vector so now that we know that force is a vector we can eliminate answer choice e what else do we need to know you need to know that distance is scalar but displacement is a vector if you were to say a person ran 45 meters you're describing a person's distance because you didn't apply direction to it but if you were to say a person ran 45 meters east you're now describing the displacement of the person and not his distance so displacement is basically distance with direction so we can eliminate answer choice a now you also need to know that speed is a square a scalar quantity and a velocity is a vector quantity so just as displacement is distance with direction velocity is speed with direction velocity tells you how fast you're going and where you're going speed simply tells you how fast you're going and by the way displacement is the change in position acceleration tells you how fast the velocity is changing acceleration is also a vector so for this problem the correct answer is answer choice d mass is a scalar quantity for instance let's say an object has a mass of 10 kilograms you won't say that the object has a mass of 10 kilograms north it wouldn't make sense so mass is a scalar quantity it has magnitude only but no direction now consider this problem a force vector has a magnitude of 100 newtons directed at an angle of 30 degrees above the x axis calculate the magnitude of the x and y components of this force vector so first let's write some equations so let's say we have the force vector f we can break it up into its x component and its y component let's call this angle theta now let's review some things that you might have learned in trigonometry if you've taken that class there's something called sohcahtoa let's focus on the so part of silchator the s stands for sine sine of the angle is equal to the o represents the opposite side h is the hypotenuse so opposite to the angle theta is f y so let's write opposite adjacent to the angle theta is f x and then across the 90 degree angle that is the hypotenuse which is f in this case so using the formula sine theta is opposite over hypotenuse we can say that sine theta is going to be f y over f now if you were to rearrange this equation if you were to multiply both sides by f you'll get that the y component of the force vector is the magnitude of f times sine of the angle theta with respect to the x-axis that's the first equation that you need to be aware of now let's consider the second equation in the cop part of circa tour so c stands for cosine cosine of the angle is going to equal the adjacent side which is f x over the hypotenuse which we know it to be f now if we do the same thing if we were to multiply both sides by f we'll get that the x component of the force vector is the magnitude of f times cosine of the angle now for the last one toa tangent theta is equal to o or opposite which is f y over the adjacent part which is f x so I like to use this formula to calculate the angle because in some problems you need to determine the magnitude and the angle to determine the angle you need to take the arc tangent or the inverse tangent if we were to take the arctan of both sides of the equation the arctan and the tan will cancel on the left so we'll get that the angle is equal to arctan fy over fx so these are some things you want to write down because it's going to be very helpful particularly when you're solving problems later in this video now let's go back to a right triangle let's say this is a b and c according to the pythagorean theorem we know that c squared is equal to a squared plus b squared well c is the hypotenuse so we could replace c with f a corresponds to f of x in this example and b corresponds to f of y so if we wish to calculate the magnitude of a vector and we know the x and y components it's simply going to be the square root of f sub x squared plus f sub y squared so make sure you're familiar with these  four formulas now let's go ahead and finish this problem so first let's draw a picture so we have a force vector that is directed at an angle of 30 degrees above the x-axis so here we have the x-axis and this is the y-axis so it would be somewhere in that area and the magnitude of this force vector is a hundred newtons so let's break it up into its x and y components so this is going to be the x component of the vector and this is the y component of that vector and then we have our angle here which is 30 degrees so with this information go ahead and calculate the magnitude of the x and the y components of this force vector so we know that f of x is equal to f cosine theta f is a hundred and theta is 30. now cosine of 30 degrees that's equal to the square root of three over two so we have a hundred divided by two which is fifty so the exact answer is 50 square root 3. now for those of you who want a decimal value if you multiply 50 and the square root of 3 you're going to get 86.6 newtons so that's the value of f sub x we'll write as 86.6 newtons now let's do the same for the y component this is going to be f times sine theta so f is 100 and then times sine of 30. sine 30 is one half so half of a hundred is 50. so we could say that f sub y is equal to 50 degrees i mean not 50 degrees but 50 newtons that's the the unit of force now we're told to express the answer using standard unit vectors but you might be wondering what is a unit vector a unit vector is simply a vector with a left or magnitude of one now we want to express it using the unit vectors i j and k so let's draw a three-dimensional coordinate system where this is z this is x and this is y i is a unit vector along the x-axis so it has a length of one j is the unit vector along the y-axis and z I mean k is a unit vector along the z axis so we need to know is that the unit vector i is associated with the x component the unit vector j is associated with the y component and the unit vector k is associated with the z component so to express the answer using standard unit vectors we can say that the original force vector f is equal to 86.6 times the unit vector I because that's the x component and then plus 50 times j which tells us that the magnitude of the y component is 50. so this is one way in which we can express the force vector so you can express a force vector or any vector using the magnitude and the angle or you can express it in component form using the x and y components