in this lecture we are going to describe the concept of force vector and the type of operations that we often need to perform on such vectors when dealing with structures consider these structural systems they need to carry certain loads but how do we know if they are strong enough to actually carry the load safely in order to ensure the safety and reliability of such systems we need to be able to determine stresses that develop in the structural members under the applied loads we often define stress in terms of force therefore the concept of force is central to the study of the mechanics of structures force is a vector it has a magnitude and a direction consider the weight of a typical traffic light for structural analysis purposes we can represent it as a vector like this a vector as a tail ahead and a magnitude the length of the vector represents its magnitude alternatively we can write the vectors magnitude next to it like this a vector also has a direction here the forces pointing downward or more accurately it makes a 90 degree angle in the clockwise direction with the horizontal axis we can also define a line of action for our vector this is the line that passes through the head and tail of the vector and extends infinitely in either direction when dealing with mechanics problems we may also be interested in the point of application of the force what kind of operations do we need to be able to perform on force vectors consider this construction crane each of the cables attached to the tip of the mast carries a force like this collectively these forces are pressing down on the mast therefore the force that the mast must be able to carry is the sum of the cable forces finding this sum requires a vector addition an operation on vectors that we are going to examine in this lecture here is a force vector having a magnitude of 25 kilonewtons the vector makes eight thirty six point 87 degree angle with x-axis since the vector lies in an XY plane it can be represented using a vector along x-axis and a vector along Y axis like this we often want to represent vectors in terms of their x and y components because it makes a vector addition simpler but how do we go from here to here how do we determine the magnitude of these two component vectors a vector and its x and y components always form a right triangle like this the length of each side of the triangle represents the magnitude of a vector this length represents the magnitude of our 25 kilonewton force this length represents the magnitude of the X component of the force 20 kilonewtons and this length represents the magnitude of the Y component of the force which is 15 kilonewtons note how the Pythagorean theorem holds true here alternatively instead of shifting the Y component of the force we can shift its X component in order to form a right triangle like this it gives us the exact same relationship between the sides of the triangle as before given that this is always a right triangle and we know it's interior angles then we can use trigonometry to figure out the side lengths that is knowing the magnitude and orientation of a force vector we can easily determine its x and y components let's refer to this unknown length as FX and call this length FY then we can write sine of thirty six point 87 equals F Y over 25 and cosine of thirty six point 87 equals FX over twenty five this gives us 15 kilonewtons for FY and 20 kilonewtons for FX so we can replace this vector with these component vectors and vice versa now consider this force vector here is the line of action of the vector it is important to note that no matter where the vector is located on its line of action we get the same x and y components for the force to calculate the vector components we place the origin of our coordinate system at the tail of the vector let's assume the vector makes a 45 degree angle with a horizontal axis we construct our right triangle by making our vector its hypotenuse we label the horizontal side of the triangle FX and we call the vertical side F Y now we use the definition of sine and cosine to determine these unknown lengths these are the X&Y components of the given force vector what if the vector has a negative magnitude how do we determine its components suppose our vector has a magnitude of negative 10 kilonewtons and it makes a 30 degree angle with y-axis a negative magnitude simply means the vector is actually acting in the opposite direction to what is shown one way to deal with such a vector is to correct its direction to rotate it 180 degrees and change its magnitude from negative to positive like this now we can determine its x and y components as before let's place the tail of the vector at the origin here is our right triangle in which the 10 kilonewton force acts as the hypotenuse label the length of the horizontal side FX and refer to the vertical side as FY then solve for the side lengths using the definition of sine and cosine like this here are the vector components now suppose we were given the x and y components of a vector and we are asked to determine its magnitude and direction here all we have to do is to add the two component vectors in order to determine the magnitude of the parent vector we start by forming a right triangle by moving one of the vectors without changing its orientation such that its tail is placed at the head of the other vector this can be done in one of two ways either it like this or like this we then draw a tail to head vector like this we call this the resultant vector we are after its magnitude and direction note that the vector defines the hypotenuse of the triangle therefore we can calculate its length using the Pythagorean theorem what remains to be determined is the orientation of the resultant vector which we can define using the angle that the vector makes with the horizontal or vertical axis the angle can be calculated using the definition of tangent like this so this pair of component vectors is equivalent to this resultant vector knowing how to express a vector in terms of its x and y components we are now ready to see how two or more vectors can be added together consider the following two vectors to add them together let's consider each vector in isolation and determine its X&Y components like this you we can replace the original vectors with these two pairs of vectors we then add these X components together this gives us a force vector along x axis with the magnitude of 7.3 kilonewtons and we add the Y components together to get a vector with a magnitude of 7.7 kilonewtons this represents the sum of these vectors in component form we then add the two components together to get a single resultant vector the magnitude of this vector can be determined using the Pythagorean theorem like this and the orientation of the vector can be determined using trigonometry like this here is our resultant vector in summary vector addition involves representing each vector in terms of its X&Y components then add all the X components together and add all the Y components together in order to determine the X and y components of the resultant vector see if you can correctly solve the following vector addition and subtraction problems you