Given: u <-1, 7> and v <4, -3> Find: (u • v)(u + v)
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<25, 100>
<-25, 100>
<-75, -100>
<75, -100>
To find the solution to this problem, we first need to understand what the dot product and vector addition are.
The dot product (u • v) of two vectors u and v is a scalar that measures the extent to which the two vectors are aligned. It is calculated by taking the product of their corresponding components and then summing up the results. In other words, (u • v) = u1*v1 + u2*v2, where u1 and u2 are the components of vector u, and v1 and v2 are the components of vector v.
Vector addition (u + v) is simply combining the corresponding components of two vectors to form a new vector. In this case, (u + v) = (u1 + v1, u2 + v2).
Now, let's calculate the given expression (u • v)(u + v):
Step 1: Calculate (u • v)
(u • v) = u1*v1 + u2*v2
(u • v) = (-1)*(4) + (7)*(-3)
(u • v) = -4 - 21
(u • v) = -25
Step 2: Calculate (u + v)
(u + v) = (u1 + v1, u2 + v2)
(u + v) = (-1 + 4, 7 + (-3))
(u + v) = (3, 4)
Step 3: Calculate (u • v)(u + v)
(u • v)(u + v) = -25 * (3, 4)
(u • v)(u + v) = (-25*3, -25*4)
(u • v)(u + v) = (-75, -100)
Therefore, the answer is < -75, -100 >.