# Given the vectors u-v=3i+2j and u+v=2i+3j, what is the difference u^2-v^2?

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We have u-v=3i+2j and u+v=2i+3j, and we need to determine u^2 - v^2.

u^2 - v^2 = (u - v)(u + v)

=> (3i + 2j)(2i + 3j)

=> 6i^2 + 9i*j + 4j*i + 6j^2

i^2 = 1 and j^2 = 1 , i*j = j*i = 0

=> 6*1 + 9*0 + 4*0 + 6*1

=> 12

**The required value for u^2 - v^2 = 12**

Since u^2 - v^2 is the difference of 2 squares, we'll use the identity:

u^2 - v^2 = (u - v)(u + v)

We'll have, form enunciation, u-v =3i+2j and u+v=2i+3j.

(u - v)(u + v) = (3i+2j)(2i+3j)

We'll remove the brackets:

(3i+2j)(2i+3j) = 6i^2 + 9i*j + 4i*j + 6j^2

We know that i^2 = 1 and j^2 = 1 , i*j = 0.

(3i+2j)(2i+3j) = 6 + 6 = 12

**The value of the difference of 2 squares is u^2 - v^2 = 12.**