For vector A=2ax + 5ay + 4az,B=3ax + ay + 5az,C=ax -6a.calculate(a)A.B (b)A*B (c)A .(B-C) (d)A*(B+C)?

Here we go:





(a) This is a straight forward dot product with the twist the constants involve the constant a.


A.B = (A1)(B1) + (A2)(B2) + (A3)(B3)





where:


A1 = 2a


A2 = 5a


A3 = 4a


B1 = 3a


B2 = a


B3 = 5a





A.B = (2a)(3a) + (5a)(a) + (4a)(5a)


A.B = (6a^2) + (5a^2) + (20a^2)


A.B = 31a^2





(b) and the cross product


A*B = (A2B3−A3B2)x+(A3B1−A1B3)y+(A1B2−A2B1)z





so:





A*B = ((5a)(5a) − (4a)(a))x + ((4a)(3a) − (2a)(5a))y + ((2a)(a) − (5a)(3a))z





A*B = (25a^2 − 5a^2)x + (12a^2− 10a^2)y + (2a^2 − 15a^2)z





A*B= (20a^2)x + (2a^2)y − (13a^2)z





(c) subtract then dot


B - C = 3ax + ay + 5az - ax + 6a


B - C = 2ax + ay + 5az + 6a





A.(B-C) = (2a)(2a) + (5a)(a) + (4a)(5a) + 6a


A.(B-C) = (4a^2) + (5a^2) + (20a^2) + 6a


A.(B-C) =29a^2 + 6a





(d) add then cross


B + C = 3ax + ay + 5az + ax - 6a


B + C = 4ax + ay + 5az - 6a





A*(B+C) = ((5a)(5a) − (4a)(a))x + ((4a)(4a) − (2a)(5a))y + ((2a)(a) − (5a)(4a))z - 6a





A*(B+C) = (25a^2 − 5a^2)x + (16a^2− 10a^2)y + (2a^2 − 20a^2)z - 6a





A*(B+C) = (20a^2)x + (6a^2)y − (18a^2)z - 6a





Hope that helps. Let me know if you have any questions.

For vector A=2ax + 5ay + 4az,B=3ax + ay + 5az,C=ax -6a.calculate(a)A.B (b)A*B (c)A .(B-C) (d)A*(B+C)?
b, use b it's confusing using 5 ay