vector calculation?

Question

vectors a, b and c are of magnitudes 2,2, and 3 respectively. the angle between b and c is π/3, between c and a 2π/3, and between a and b it is π/4.

Calculate
lla + b + cll

and

lla x (a + b)ll

Answer 1

Calculate lla + b + cll.

lla + b + cll² = •

• = a² + b² + c² + 2ab + 2ac + 2bc

= ||a||² + ||b||² + ||c||²
+ 2||a|| ||b|| cos(π/4) + 2||a|| ||c|| cos(2π/3) + 2||b|| ||c|| cos(π/3)

= 2² + 2² + 3² + 2*2*2*(1/√2) + 2*2*3*(-1/2) + 2*2*3*(1/2)

= 4 + 4 + 9 + 8/√2 - 12/2 + 12/2 = 17+ 8/√2

lla + b + cll = √(17+ 8/√2) ≈ 4.7599217
_____________________

Calculate ll a X (a + b) ll.

The cross product is distributive over addition.

ll a X (a + b) ll = || aXa + aXb || = || 0 + aXb || = || aXb ||

= || ||a|| ||b|| sin(π/4) || = 2*2*(1/√2) = 2√2 ≈ 2.8284271

Answer 2

Previous poster erroneously assumed these were all in a plane.

For the first part, just take (a + b + c) * (a + b + c)

Expand that via the distributive law in the usual way. Remember the relationship between dot products, magnitudes, and the cosine of the angle between the vectors.

And don't forget to take the square root at the end to get the magnitude. :)

Answer 3

No possible permutation complies with the angles provided