Compute the cross product u×v for u=(3,-2,5) and v=(1,0,-4). Which vector is correct?

Compute the cross product u×v for u=(3,-2,5) and v=(1,0,-4). Which vector is correct?

• A. (-8, -17, -2)
• B. (8, 17, 2)
• C. (8, -17, 2)
• D. (0, 0, 0)

Answer: (B)

The cross product u×v gives a vector perpendicular to both u and v, with magnitude equal to the area of the parallelogram spanned by u and v, and its direction is determined by the right-hand rule. Using the formula u×v = (u2 v3 − u3 v2, u3 v1 − u1 v3, u1 v2 − u2 v1) for u = (3, −2, 5) and v = (1, 0, −4): - First component: (−2)(−4) − (5)(0) = 8 − 0 = 8 - Second component: (5)(1) − (3)(−4) = 5 + 12 = 17 - Third component: (3)(0) − (−2)(1) = 0 + 2 = 2 So u×v = (8, 17, 2). The other options don’t match these components (they would have signs flipped or include zero where not appropriate), so this one is the correct cross product.

The cross product u×v gives a vector perpendicular to both u and v, with magnitude equal to the area of the parallelogram spanned by u and v, and its direction is determined by the right-hand rule.

Using the formula u×v = (u2 v3 − u3 v2, u3 v1 − u1 v3, u1 v2 − u2 v1) for u = (3, −2, 5) and v = (1, 0, −4):

• First component: (−2)(−4) − (5)(0) = 8 − 0 = 8

• Second component: (5)(1) − (3)(−4) = 5 + 12 = 17

• Third component: (3)(0) − (−2)(1) = 0 + 2 = 2

So u×v = (8, 17, 2). The other options don’t match these components (they would have signs flipped or include zero where not appropriate), so this one is the correct cross product.