`|vecA|=3`, `|vecB|=5` and the angle formed by `vecA` and `vecB` is 30 degrees. What is `|vecA+vecB|`?

txmedteach | Certified Educator

This question is simply trying to determine how to add two vectors without being given a coordinate plane. Believe it or not, though, the lack of a given axes helps us because we can set our own axes based on one of the vectors!

So, let's just go ahead and say `vecA` is on the x-axis. This means that vecA will have the following Cartesian coordinates:

`vecA = (3,0)`

We know that `vecB` is at an angle of 30 degrees to `vecA`. It does not matter for this problem whether that means `vecB` goes into the first or fourth quadrant (or even off the page!). Thus, we will say `vecB` goes into quadrant I at an angle of 30 degrees from `vecA`.

Now that we have established our directions, let's find the Cartesian coordinates for `vecB`. We know the magnitude and direction for `vecB` , so we can calculate the coordinates in the following way:

`vecB = (|vecB|costheta, |vecB|sintheta)`

Here, `theta` will be 30 degrees because of the 30 degree difference between `vecB` and `vecA` and because `vecA` is on the x-axis. Therefore, we can make the following substitutions:

`vecB = (5cos(30^o), 5sin(30^o))`

Evaluating the trigonometric functions and simplifying:

`vecB = (5/2, 5sqrt3/2)`

Now we have our two vectors, and we can now perform vector addition. All we do here is add the corresponding Cartesian coordinates:

`vecA + vecB = (3 + 5/2, 0 + 5sqrt3/2) = (11/2,5sqrt3/2)`

All that is left to do is find the magnitude of this resulting vector:

`|vecA+vecB|=sqrt((11/2)^2 + (5sqrt3/2)^2) = sqrt(121/4+75/4)=sqrt(196/4) = sqrt(49) = 7`

Therefore, our final magnitude is 7.