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Adding 2 1 D Vectors :
- If both are in the same dimension, combine magnitudes, and pay attention to minus signs > If both vectors are in the same direction, you add the magnitudes and keep the same sign; if they are in the opposite direction, you subtract and take the sign of the "larger" magnitude.
- If they are in different directions (say, one is horizontal and the other is vertical) we have to use geometry to determine the magnitude of the sum; we then use trigonometry to determine the direction (expressed as an angle).
Example: A student walks 300 meters East, and 400 meters North. What is her displacement? Express both Magnitude and Direction.
Vector Components: Assume you have a vector (say, a velocity, v ) that is directed in two dimensions at once; perhaps it is directed up and to the right, such that it has a magnitude of 20 m/s and is directed at an angle of 30 degrees. How can we analyze this kind of motion? How can we deal with a velocity (or any vector) that is in more than one dimension? We will try to take this single vector in 2 dimensions, and express it as a sum of 2 vectors, each of which is only in one direction. We call these 2 vector the "vector components ", one of which is in the x-direction (the x-component ), and the other in the y-direction (the y- component ). To find the x- and y-components, we need to use a little trig. The x-component of the velocity v is called vx , and we can calculate it by the following expression: vx = vCosθ [ degree mode! ] The y-component of the velocity v is called vy , and we can calculate it by the following expression: vy = vSinθ [ degree mode! ] *Note that, if you were to add vx to vy , the vector sum would yield the vector v!
Vector Addition - by Components: Assume we have 2 separate displacements given by vectors A and B , such that A = 100 m @ 53 degrees and B = 80 m @ 150 degrees We want to know the sum of vectors A and B , which we will call R. We ultimately want to state vector R in terms of its Magnitude and Direction. [You may ask: is R just 180 m @ 203 degrees? The answer is no; you cannot just add the magnitudes and directions of A and B. The process is a bit more involved than that.] To determine the Magnitude of R , we first need to take vectors A and B and Decompose them into their x- and y-components.
- To do this, recall that Ax = ACosθ , and Ay = ASinθ. We use a similar approach for Bx and By. Ax = ACosθ = 100 Cos( 53 ) = 60 Ay = ASinθ = 100 Sin( 53 ) = 80 Bx = BCosθ = 80 Cos( 150 ) = - 69. 3 By = BSinθ = 80 Sin( 150 ) = 40
