magnitude and direction of resultant

[josh90]

Find the magnitude and direction of the resultant for vectors y = (3, 2) and z = (-2, 3). Give the magnitude as an exact answer (a whole number or a radical), and round the direction to the nearest tenth.

So far I found the resultant of those two vectors, which is (1, 5). But I am having problems finding the magnitude and direction of the resultant. My book only showed me how to find the magnitude and direction of two vectors, not one resultant. Can anyone please explain it to me? Thank you!

 

[pka]

So far I found the resultant of those two vectors,which is (1,5).
My book only showed me how to find the magnitude and dir. of two vectors not one resultant?

Your resultant vector is correct. But I find it hard to think that your book does not give you the other definitions.

If \(\displaystyle v = < a,b > \quad \Rightarrow \quad \left\| v \right\| = \sqrt {a^2 + b^2 } :\) that is magnitude.

Now unfortunately, ‘direction’ has many different meanings, each depends upon the particular text in use. In general, there is an idea of direction angles: \(\displaystyle - \pi < \alpha \le \pi\) is the angle the vector makes with the positive x-axis and \(\displaystyle - \pi < \beta \le \pi\) is the angle made with the positive y-axis.

 

[josh90]

Geometry

What I mean by direction is lets say I have vector PQ P(3,8) and Q(-4,2).

now to get the direction I would do 2-8/-4-3 and I would get 6/7 then I would do measure angle P=tan-1 6/7=40.6 degrees which is the direction, but now how would I do that with just the resultant <1,5>. And for the magnitude I got 5 is that correct? Thx for your help

 

[pka]

If P(3,8) and Q(-4,2) are two points, then the vector PQ=<-4-3,2-8>=<-7,-6>.
The angle that vector makes with the positive x-axis is \(\displaystyle \arctan \left( {\frac{6}{7}} \right) - \pi.\) Some textbooks want the absolute of that number. As I have explained above, the text may want something else altogether. You simply must read your text for the definitions in regard to direction. This is no standard here, in fact textbooks rarely agree on this one.

BTW:\(\displaystyle \left\| { < - 7, - 6 > } \right\| = \sqrt {49 + 36} .\)