Adding Vectors and Vector Components
October 02, 2013
Adding 2 1D 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).
Honors Physics
Adding Vectors and Vector Components
October 02, 2013
Example: A student walks 300 meters East, and 400 meters North.
What is her displacement? Express both Magnitude and Direction.
Honors Physics
Adding Vectors and Vector Components
October 02, 2013
Example: A car drives 20 miles West, and then 25 miles North. What
is its displacement, in terms of magnitude and direction?
Honors Physics
Adding Vectors and Vector Components
October 02, 2013
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 ycomponent). 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!
Honors Physics
Adding Vectors and Vector Components
October 02, 2013
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.
1. 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
Honors Physics
Adding Vectors and Vector Components
October 02, 2013
2. Next, we can determine the x- and y-components of vector R by
adding the x- and y-components of A and B:
Rx = Ax + Bx = 60 + (-69.3) = -9.3
Ry = Ay + By = 80 + 40 = 120
3. To determine the Magnitude of R, we apply the Pythagorean Theorem:
R = √Rx2 + Ry2 = √(-9.3)2 + (120)2 = 120.4
4. To determine the Direction of R, we will use the "arc-Tan" trig function,
using the "2nd Tan" function on the calculator:
θ = Tan-1(Ry/Rx) = Tan-1(120/-9.3) = -85.6 degrees
**Here we need to be careful... what exactly does that angle mean? In
order to make sense of this angle, look at our diagram above. Vector R is
in the 2nd quadrant. The Rx is negative, and Ry is positive. In a case
such as this (2nd quadrant), we will Add 180 degrees to the angle given;
this gives us a final answer of θ = -85.6 + 180 = 94.4 degrees.
Honors Physics
Adding Vectors and Vector Components
Honors Physics
October 02, 2013