more vectors?

Question

i had 100 probs 2 do i did all but these 16

7. identify the decomposition of u into the sum of two orthogonal vectors. One of which is the projection of v onto u
u = {2,-1} v= {1,-5}

8. find zw
z= 8(cos pi/6 + i sin pi/6) w= 4(con pi/12 + i sin pi/12)

I think its 32(cos pi/4 + i sin pi/3)

9. find the direction angle of the vector
v= -3i -10j

10. identify the trigonometric form of the complex number
3 – 8i

11. find the number by which the components of the vector can be divided to find the unit vector in the same direction
u= -3i + 3j
12. a box being pulled by two ropes . one rope exerts 55lbs and force of 24 degrees the other rope exerts 95 lbs and a force of 23 degrees. Find the direction and magnitude of the resultant of two forces

13. find the value of the expression:
(cos30 degrees + i sin 30 degrees) / (cos 75 degrees + i sin 75 degrees)

Answer 1

7. I believe what you mean is: Identify the decomposition of u into the sum of two orthogonal vectors. One of which is the projection of u onto v (instead of v onto u, since u is to be decomposed).
u = {2,-1} v= {1,-5}
Let us write one component as u// and the other as u.
| u// | = u•v/|v| = (2+5)/sqrt(1+25) = 7/sqrt(26), hence:
u// = | u// |*v/|v| = {7/26, -35/26}
Since u// + u = u, we have:
u = u - u// = {2-7/26, -1+35/26} = {45/26, 9/26}

8. Do you mean "w= 4(cos pi/12 + i sin pi/12)" instead of "w= 4(con pi/12 + i sin pi/12)"
No, it is not 32(cos pi/4 + i sin pi/3)
It is: 32(cos pi/4 + i sin pi/4)

9. tan^-1(10/3)

10. sqrt(3^2+8^2) = sqrt(73)
Let θ = 2*pi - acos(3/sqrt(73))
The answer is: sqrt(73)*(cos θ + i sin θ)

11. 3*sqrt(2)

12. Do not understand your problem.

13. (cos30° + i sin 30°) / (cos 75° + i sin 75°)
= cos (-45°) + i sin (-45°)
= (1 - i)*sqrt(2)/2

Answer 2

For the complex number problems, check out:
http://en.wikipedia.org/wiki/Complex_number

For #8, both angles are pi/4

For problem #7 you need to find:
- a vector w which is orthogonal (i.e. perpendicular) to v
- two numbers, a and b, such that u = av + bw

To find w, just rotate v by 90 degrees. That means x -> y, y-> -x or {x,y} -> {-y, x}

To find a and b, recall that if X is any vector and Z a unit vector, then the magnitude of the component of X in the direction Z is given by the dot product of X and Z. If Z is not a unit vector, then you have to scale appropriately by the magnitude of Z. This is what is going on in #11

That is (1/|Z|)Z is a unit vector in the same direction as Z.

#12 makes no sense as written. I assume it meant to say a force of 55 lbs at an angle of 24 degrees, etc. That is specifying the vector in terms of angle and magnitude rather than components (i.e. rectangular coordinates)

To add the two, you need to convert both to rectangular coordinates, do the sum, and convert back.