To find the specified scalar, we need to first find the dot product of U and V, and then the dot product of U and W, and add these two dot products together:
U·V + U·W
where U·V denotes the dot product of U and V, and U·W denotes the dot product of U and W.
The dot product of two vectors a = (a1, a2) and b = (b1, b2) is given by:
a · b = a1b1 + a2b2
Using this formula, we can find the dot product of U and V:
U · V = (4i - j) · (5i + j) = 4i · 5i + 4i · j - j · 5i - j · j = 20i^2 + (4i · j - 5i · j) - j^2 = 20 + (-i · j) - 1 = 19 - i · j
Next, we can find the dot product of U and W:
U · W = (4i - j) · (i + 6j) = 4i · i + 4i · 6j - j · i - j · 6j = 4i^2 + (4i · 6j - j · i) - 6j^2 = 4 + (24i · j) - 6 = -2 + 24i · j
Now we can substitute these values into the original expression:
U·V + U·W = (19 - i·j) + (-2 + 24i·j) = 17 + 23i·j
Therefore, the specified scalar is 17 + 23i·j.