Vector

(a) (p + q)*(p + q) = p * p + 2p*q + q*q * means dot product.
= 5^2 + 2(5)(7) cos 120 + 7^2
= 25 - 70 cos 60 + 49
= 74 - 35 = 39.
so abs(p + q) = sqrt 39.
(b) (q - r)*(q - r) = q*q - 2q*r + r*r
= 7^2 - 2(7)(3)cos 60 + 3^2
= 49 - 21 + 9 = 37
so abs(q - r) = sqrt 37.
(c) r = - 3p/5, so p - q + r = p - q - 3p/5 = 2p/5 - q
so ( p - q + r)*(p - q + r) = (2p/5 - q)*(2p/5 - q)
= (4/25)p*p - (4/5)p*q + q*q
= (4/25)(25) - (4/5)(5)(7) cos 120 + 49
= 4 + 14 + 49 = 67
so abs(p - q + r) = sqrt 67.