How do you add 2D vectors a+b with the component method?

Question

How do you add 2D vectors a+b

• if the magnitude of vector a is 10, its angle with (+) x-axis is 45 and angle with (+) y-axis is 45 ...
• then vector b has a magnitude of 3 angle with (+) x-axis of 30 and angle with (+) y-axis of 120

Answer 1

Vectors have components. You can think of them as being coordinates. 

 

Here's a vector: (a, b)

 

Here's another vector: (c, d)

 

If I want to add these two together, I add their components: (a + c, b + d).

 

I'm not sure why they're calling it the "component method". That sounds weird and intimidating. And I don't know if this isn't the only method. If it is, why distinguish it by calling it something like that?

 

Anyway, so you can add those vectors if you know what they are. So what we have to do is get you to write them first. 

 

You say the first one has a magnitude of 10 and is at a 45^o to the positive x and y axes.

 

Imagine I drew a line segment that made that angle to those axes. One endpoint is at the origin and the other is a coordinate I don't know yet. Then imagine that I dropped a line from that coordinate to the x-axis in such a way that it made a right angle with it. This would give you a right triangle. 

 

This would give you a right triangle with a hypotenuse of 10 and two 45^o angles. Let's call this coordinate we're trying to figure out (x, y). Notice that on this triangle, the y is the length of the vertical leg and the x is the length of the horizontal leg. Looking at this triangle, you can see that sin(45^o) = y/10 and that cos(45^o) = x/10. You can use this info to find x and y. (x, y) = (10cos(45^o), 10sin(45^o)).

 

This is true for any vector you draw at a given angle in the coordinate plane: (x, y) = (rcos(θ), rsin(θ)) where r is the length of the vector (and the radius of a right triangle you're able to draw from it).

 

So far, we've found one of those two vectors. We still have to find the other. Be careful when you do this. You want to get the correct angle. If a vector makes a 30^o with the positive x axis but a 120^o with the positive y axis, it means your angle must be -30^o (remember that's thirty degrees clockwise or the same thing as 330^o). 

 

To find this other vector, we know it has length 3 and angle 330^o. This means we'll find it to be (3cos(330^o), 3sin(330^o)).

 

When we add these vectors together, it looks like

 

(10cos(45^o), 10sin(45^o)) + (3cos(330^o), 3sin(330^o))

 

= (10cos(45^o) + 3cos(330^o), 10sin(45^o) + 3sin(330^o)).

 

 

To go from there, recall that sin(45^o) = cos (45^o) = squareroot of 2 over 2, that cos(330^o) = square root of 3 over 2, and that sin(330^o) = -1/2. 

 

And because I can't type this stuff well that is all I will write. 

 

Answer 2

Vector addition isn't as horrible as it seems, though it requires some knowledge of trigonometry.

 

For a 2 dimensional vector, whenever we're given a pair of vectors a = <x₁, y₁> and b = <x₂, y₂>, if we're asked to add the vectors, we simply add the x components and the y components.

 

a + b = <x₁ + x₂, y₁ + y₂>

 

In this case, you weren't given an <x,y> format for your vector, so you have to use some trig to figure out the x and y components. We do this by treating the problem as if we're trying to convert from polar coordinates (r, theta) to cartesian coordinates (x, y)

 

x = magnitude * cos(angle)

y = magnitude * sin(angle)

 

In this case, you have to measure the angle from the x axis positive (which thankfully the problems listed here are!)

 

So for vector a, we're given the magnitude is 10 and the angle is 45 degrees

 

so,

x_a = 10 * cos(45) = 10 * sqrt(2) / 2 = 5 * sqrt(2)

y_a = 10 * sin(45) = 10 * sqrt(2) / 2 = 5 * sqrt(2)

 

(sqrt means square root, by the way)

 

Now we look at vector b, we're given the magnitude of 3 and an angle of 30 degrees above the x axis.

 

x_b = 3 * cos(30) = 3 * sqrt(3) / 2 = 3/2 * sqrt(3)

y_b = 3 * sin(30) = 3 * 1/2 =           3/2

 

Now we simply add the two component vectors to get the new vector of a + b:

 

a + b = <5sqrt(2), 5sqrt(2)> + <3/2 * sqrt(3), 3/2>

 

a + b = <5 * sqrt(2) + 3/2 * sqrt(3), 5 * sqrt(2) + 3/2>

 

if we're turning this into decimals, we get:

 

a + b = <9.67, 8.57>