4.1     a) 8*6*cos(2*pi/3)
        b) 1/2*(4^2-3^2-2^2)
	c) -1/2*(10^2-12^2-4^2)
        d) 3*1*cos(pi)
        e) 2*4-3*2
        f) 2*4-3*2-5*1

4.2     {3, 1, -2}.{1, -4, -5}

4.3     -2*(4*sqrt(2))^2+3*4*sqrt(2)*8*cos(pi/4)-8^2

4.4     sqrt(4^2+2^2+6^2+2*(2*4+4*6+2*6)*cos(pi/3))

4.5     sqrt(3^2+9*2^2-6*3*2*cos(pi/3))

4.6     sqrt(4^2+7^2+2*4*7*cos(pi-2*pi/3))

4.7     {2, 1, 1}.sqrt(2){1, -1, -1}, sqrt(2^2+1^2+1^2) - sqrt(sqrt(2)^2+sqrt(2)^2+sqrt(2)^2)

4.8     a) arccos({3, -1}.{2, 4}/sqrt((3^2+1^2)(2^2+4^2)))
        b) arccos({0, 1, -1}.{0, -1, 0}/sqrt((0^2+1^2+(-1)^2)(0^2+(-1)^2+0^2)))

4.9     arccos({3, 0, 4}.{7, 0, 1}/sqrt((3^2+0^2+4^2)(7^2+0^2+1^2)))

4.10    a) arccos({2, -7}.{5, 2}/sqrt((2^2+(-7)^2)(5^2+2^2))))
        b) arccos({4, 16, -12}.{-6, -24, 18}/sqrt((4^2+16^2+(-12)^2)((-6)^2+(-24)^2+18^2))))
        c) arccos({2, 1, -8}.{7, 2, 2}/sqrt((2^2+1^2+(-8)^2)(7^2+2^2+2^2))))
        d) arccos({5, -3, 2}.{1, 3, 4}/sqrt((5^2+(-3)^2+2^2)(1^2+3^2+4^2))))

4.11    arccos((8-5)/6)
     
4.12    a) {2, -5}.{3, 4}/sqrt(3^2+4^2) // project {2,-5} onto {3,4}
        b) {1,-4, 0}.{0, -2, 1}/sqrt(0^2+(-2)^2+1^2) // project {1,-4, 0} onto {0, -2, 1}
        c) {2, 1,-3}.{7, -2, 4}/sqrt(7^2+(-2)^2+4^2) // project {2, 1,-3} onto {7, -2, 4}