How Do You Calculate Vector Components and Angles in 3D Space?

[Neophyte]

Homework Statement

What are
(a) the x component,
(b) the y component, and
(c) the z component of
r= a-b +c if
a = 6.5i + 4.2j - 6.7k,
b = -6.3i + 1.5j + 1.4k, and
c = 8.2i + 4.3j + 5.9k.
(d) Calculate the angle between r and the positive z axis.
(e) What is the component of a along the direction of b?
(f) What is the magnitude of the component of a perpendicular to the direction of b but in the plane of a and b and ?

Homework Equations

a . b = abcosθ
c = absinθ
tanθ= (aj/ai)

The Attempt at a Solution

a) 21
b) 7
c) -2.2

21i + 7j - 2.2k

d)√(441+ 49 + 4.84) = 22.24

θ=sin-1 = 7/22.24
θ = 18.34 /*Incorrect I am pretty sure ;/ */
I took y/r r as hypot and y as opposite

e)Well I know how to find the ay = asinθ etc but the z I do not know.
So I tried
direction of b -i,+j,+k
thus a = -6.5i + 4.2j + 6.7k but then I believe I have three components. So I am clueless.
f) close the book
a X b produces third vector c

i j k
6.5i + 4.2j - 6.7k
-6.3i +1.5j + 1.4k

= 15.93i - 33.11j + 36.21k
= magnitude 51.58

 

[rl.bhat]

d) You have to find the angle between -z-axis and r, calculate cosθ = z/r, then find (180 - θ ) to find the angle with + z-axis.

 

[Neophyte]

d) You have to find the angle between -z-axis and r, calculate cosθ = z/r, then find (180 - θ ) to find the angle with + z-axis.

so cosθ = 2.2/22.24
θ = 84.3
180 - θ = 95.6 degrees?

If - 2.2k was positive wouldn't it be the same thing as the + z-axis because cos is same for +/-? Will I always do (180 - θ)?

I had done e) wrong
(e) What is the component of a along the direction of b?
Its basically
a.b = abcosθ
a.b / b = component along direction of b

(f) What is the magnitude of the component of a perpendicular to the direction of b but in the plane of a and b and ?
What is it talking about

I thought it had something to with
aXb= c = absinθ

But then c is the direction perpendicular to the plan of a and b not in it.
So is a = a; b = c; c=a;
aXc = b = acsinθ
But I don't have the c :\\\\\\\\\\\\

 

[rl.bhat]

e) component of a along b is acosθ = abcosθ/b = a.b/b