if U=i+3j-2k and V=4i-2j-4k are vectors find U.V, 3U+V& (2U+V).(U-2V) (a)6,10,4(b)4143(c)11,4,5(d)5,11,6

not sure what the alleged answers are supposed to mean, but

3U+V = 3+4,9-2,-6-4 = 7i+7j-10k

work the other linear combinations the same way, and for the dot product multiply pairs and add.

What results do you get?

To find U · V (the dot product of vectors U and V):

1. Recall that the dot product of two vectors is given by the sum of the products of their corresponding components.

U · V = (Ui)(Vi) + (Uj)(Vj) + (Uk)(Vk)

Substituting the given values:

U · V = (i + 3j - 2k) · (4i - 2j - 4k)
= (1)(4) + (3)(-2) + (-2)(-4)
= 4 - 6 + 8
= 6

Therefore, U · V = 6.

To find 3U + V (the sum of 3 times U and V):

2. Multiply each component of U by 3 and add it to the corresponding component of V.

3U + V = 3(i + 3j - 2k) + (4i - 2j - 4k)
= 3i + 9j - 6k + 4i - 2j - 4k
= 7i + 7j - 10k

Therefore, 3U + V = 7i + 7j - 10k.

To find (2U + V) · (U - 2V) (the dot product of 2U + V and U - 2V):

3. Calculate 2U + V and U - 2V separately.

2U + V = 2(i + 3j - 2k) + (4i - 2j - 4k)
= 2i + 6j - 4k + 4i - 2j - 4k
= 6i + 4j - 8k

U - 2V = (i + 3j - 2k) - 2(4i - 2j - 4k)
= i + 3j - 2k - 8i + 4j + 8k
= -7i + 7j + 6k

4. Calculate the dot product by multiplying the corresponding components and summing them.

(2U + V) · (U - 2V) = (6i + 4j - 8k) · (-7i + 7j + 6k)
= (6)(-7) + (4)(7) + (-8)(6)
= -42 + 28 - 48
= -62

Therefore, (2U + V) · (U - 2V) = -62.

From the calculations above, none of the answer choices provided (a) 6, 10, 4; (b) 4143; (c) 11, 4, 5; (d) 5, 11, 6 are correct.