Suppose vectors a and b are vectors such that a x b =(3,1,4). What is the cross product of twice of a with twice of b?

To find the cross product of twice of vector a with twice of vector b, we can make use of the properties of the cross product.

Step 1: Multiply each component of vector a by 2 to get the "twice of a". Let's call this vector c.
c = 2 * a

Step 2: Multiply each component of vector b by 2 to get the "twice of b". Let's call this vector d.
d = 2 * b

Step 3: Calculate the cross product of vector c with vector d.
c x d

Since we know the cross product of vectors a and b, which is given as a x b =(3,1,4), we can use the property that the cross product is linear to compute the cross product of twice of a with twice of b.

Step 4: Apply the property of linearity.
c x d = 2 * (a x b)

Step 5: Plug in the known cross product of a x b.
c x d = 2 * (3,1,4)

Step 6: Compute the cross product of twice of a with twice of b.
c x d = (6, 2, 8)

Therefore, the cross product of twice of vector a with twice of vector b is (6, 2, 8).

To find the cross product of twice of a with twice of b, we need to first find the cross product of a and b, and then multiply the result by 4.

Given that a x b =(3,1,4), we can calculate the cross product of twice of a with twice of b as follows:

twice of a = 2 * a
= 2 * (a1, a2, a3)
= (2a1, 2a2, 2a3)

twice of b = 2 * b
= 2 * (b1, b2, b3)
= (2b1, 2b2, 2b3)

Now, let's find the cross product of 2a and 2b:

(2a) x (2b) = ((2a2 * 2b3) - (2a3 * 2b2), (2a3 * 2b1) - (2a1 * 2b3), (2a1 * 2b2) - (2a2 * 2b1))
= (4a2b3 - 4a3b2, 4a3b1 - 4a1b3, 4a1b2 - 4a2b1)
= (4(a2b3 - a3b2), 4(a3b1 - a1b3), 4(a1b2 - a2b1))
= (4(1 * 4 - 4 * 1), 4(4 * 3 - 3 * 4), 4(3 * 1 - 1 * 3))
= (4(4 - 4), 4(12 - 12), 4(3 - 3))
= (4(0), 4(0), 4(0))
= (0, 0, 0)

Therefore, the cross product of twice of a with twice of b is (0, 0, 0).