In the following cases compute (A x B).C
a) A = (1, 2, 0) B = (-3, 1, 0), C = (4, 9, -3)
b) A = (-3, 1, -2) B = (2, 0, 4), C = (1, 1, 1)
a) A x B = (1, 2, 0) x (-3, 1, 0) = (2*0 - 1*0, 0*0 - 1*(-3), 1*1 - 2*(-3)) = (0, 3, 7)
(A x B).C = (0, 3, 7) . (4, 9, -3) = 0*4 + 3*9 + 7*(-3) = 0 + 27 - 21 = 6
b) A x B = (-3, 1, -2) x (2, 0, 4) = (1*(-2) - (-2)*0, (-3)*4 - (-2)*2, (-3)*0 - (-2)*2) = (-2, -8, 4)
(A x B).C = (-2, -8, 4) . (1, 1, 1) = -2*1 + (-8)*1 + 4*1 = -2 -8 +4 = -6.
To compute (A x B).C, we will first find the cross product of A and B, and then take the dot product of the result with C. Let's solve the given cases step-by-step:
a) A = (1, 2, 0), B = (-3, 1, 0), C = (4, 9, -3)
Step 1: Compute the cross product of A and B.
To find the cross product, we can use the formula:
A x B = (A2*B3 - A3*B2, A3*B1 - A1*B3, A1*B2 - A2*B1)
A x B = (2*0 - 0*1, 0*(-3) - 1*0, 1*1 - 2*(-3))
= (0 - 0, 0 - 0, 1 - (-6))
= (0, 0, 7)
Step 2: Take the dot product of (A x B) with C.
The dot product can be calculated by multiplying corresponding elements and then summing them.
(A x B).C = (0*4) + (0*9) + (7*(-3))
= 0 + 0 - 21
= -21
Therefore, (A x B).C = -21.
b) A = (-3, 1, -2), B = (2, 0, 4), C = (1, 1, 1)
Step 1: Compute the cross product of A and B.
Using the cross product formula:
A x B = (A2*B3 - A3*B2, A3*B1 - A1*B3, A1*B2 - A2*B1)
A x B = (1*4 - (-2)*0, (-2)*2 - (-3)*4, (-3)*0 - 1*2)
= (4 - 0, -4 - (-12), 0 - 2)
= (4, 8, -2)
Step 2: Take the dot product of (A x B) with C.
(A x B).C = (4*1) + (8*1) + (-2*1)
= 4 + 8 - 2
= 10
Therefore, (A x B).C = 10.
In summary:
a) (A x B).C = -21
b) (A x B).C = 10