Vector direction and magnitude?

Question

Hi, I can do magnitude of vectors but I don't know how to find the direction if you asked me the following. Please, if possible, solve and explain the procedure for me. Thanks!

Find the magnitude and direction of the vectors:

u = i + j + k
v = 2i - j + 2k
w = 3i - 6j - 2k
r = -i + 3j - 2k

Thanks again

Answer 1

I've never done "direction" for a 3-D vector, but I'd assume you'd have to have two angles, i.e. express the vector using spherical coordinates. There's an explanation of spherical coordinates here (http://en.wikipedia.org/wiki/Spherical_coordinates ). θ is the same (tanˉ¹ y/x), but ρ is a second angle measured from the positive z-axis, and it is computed as cosˉ¹ z/√(x² + y² + z²). If you study the picture at Wikipedia, it should be straightforward to understand where this formula comes from.

u = i + j + k
||u|| = √(1² + 1² + 1²) = √3
θ = tanˉ¹ 1/1 = π/4
ρ = cosˉ¹ 1/√3 = 0.9553

v = 2i - j + 2k
||v|| = √(2² + (-1)² + 2²) = 3
θ = π + tanˉ¹ -1/2 = 2.6779 (add π because θ should be in quad 2 where x is negative not quad 4)
ρ = cosˉ¹ 2/3 = 0.8411

w = 3i - 6j - 2k
||w|| = √(3² + (-6)² + (-2)²) = 7
θ = tanˉ¹ -6/3 = -1.1071
ρ = cosˉ¹ -2/7 = 1.8605

r = -i + 3j - 2k
||r|| = √((-1)² + 3² + (-2)²) = √14
θ = π + tanˉ¹ 3/-1 = 1.8925 (see above for explanation about π)
ρ = cosˉ¹ -2/√14 = 2.1347

Answer 2

The question says use graphical methods so get some paper-Draw on it the x & y axes . Draw a line ,length 8 cm alongside the x axis on the left hand fringe of the y axis . placed slightly arrow on the great pointing the position its going. Now draw a line of length 9.4cm at an perspective of 37.5degrees to the x axis starting up on the great of vector B and placed slightly arrow on the great. Now connect the little arrow on the great of vector A to the starting up of vector B, which to that end, takes position to be the muse of the graph.placed slightly arrow on the great of the line This new line is the sum of the vectors A & B . degree the size and get your protractor and degree the perspective. it truly is the first bit executed . Now do the very similar back in reality this time use -B which will obviously bypass in the different route to B. In doing a majority of those issues continually keep in mind that the subsequent vector ought to continually start up from the little arrow on the great of the previous vector, and also you should hence finally end up with a loop with all the arrows pointing a similar way around the loop.