projection vectors problem

Question

1. Given p→=[−1,4,−8 ]and q→=[10,10−9], find:
2. 2p+5q→ 
3. p•q→ 
4. q×p→ 
5. p×q→ 
6. The angle between p→ and q→ 
7. projq→p→

Answer 1

a. 2p + 5q = 2[-1,4,-8] + 5[10,10,-9] = [48,58,-61]

b. p•q = -10 + 40 + 72 = 102

c. qxp = det[(i j k) (10 10 -9) (-1,4,-8)] = -44i + 89j + 50k = [-40,89,50]

d. pxq = -qxp = [40,-89,-50]

e. cosθ = p•q/(IIpII·IIqII) = 102/{(1 + 16 + 64)^1/2·(100 + 100 + 81)^1/2} = 0.67609; θ = 47.46°

f. proj_pq = (p•q)p/IIpII² = (102/81)[-1,4,-8] = [-34/27,136/27,-272/27], this is vector-projection