Question

Vectors Please Help ( What is the direction of vector A + vector B?)

What is the direction of vector A + vector B?

Answer 1

`-63.425^o`

Explanation:

Not drawn to scale

Sorry for the crudely drawn diagram but I hope it helps us see the situation better.

As you have worked out earlier in the question the vector:

`A+B=2i-4j`

in centimeters. To get the direction from the x-axis we need the angle. If we draw the vector and split it up into its components, i.e. `2.0i` and `-4.0j` you see we get a right angled triangle so the angle can be worked out using simple trigonometry. We have the opposite and the adjacent sides. From trigonometry:

`tantheta = (Opp) /( Adj) implies theta=tan^-1((Opp) /(Adj))`

In our case the side opposite the angle is `4.0cm` so `4.0cm` and the adjacent side is: `2.0cm` so:

`theta = tan^-1(4.0/2.0)=63.425^o`

Obviously this is anti-clockwise so we must put a minus in front of the angle `-> -63.425`

If the question is asking for the positive angle going clockwise around the diagram then simple subtract this from `360^o`

`-> 360-63.425=296.565^o`

Answer 2

e. `296.5^@`
f. `0^@`

Explanation:

It looks like your answer for e is wrong and perhaps you have not found an answer for f. So I will help with both.

Note: I am using the angle measuring method in which you start at the +x axis and circulate counterclockwise to the vector. So the +y axis is at `90^@` and the minus y axis is at `270^@`. Ref:
http://chortle.ccsu.edu/VectorLessons/vch05/vch05_3.html

e. From your work, `vec(A) + vec(B) = 2 " cm " hati - 4 " cm " hatj`. That puts the vector in the 4th quadrant. Draw the vector with the arrowhead at x=2, y=-4.

Let's calculate the angle `theta_e` between the -y axis and the vector. The length of the side opposite is 2 cm and the side adjacent is 4 cm.

`tan^-1(2/4) = 26.5^@`

The -y axis is already `270^@` counterclockwise from the +x axis, so the answer to e is `270^@+26.5^@ = 296.5^@`.

f. From your work, `vec(A) - vec(B) = 4 " cm " hati + 0 " cm " hatj`. Therefore the resultant lies along the x axis. That is an angle of `0^@`.

I hope this helps,
Steve