The dot product of any two vectors can be calculated as the product of their magnitudes and the cosine of the angle between them. This is represented mathematically as follows:
~v · ~w = ||~v|| * ||~w|| * cos(θ)
Where ~v and ~w are vectors, ||~v|| and ||~w|| are the magnitudes of these vectors, and θ is the angle between the vectors.
Substituting the given values into the formula, we have
~v · ~w = 3 * 5 * cos(π/3) = 7.5
Therefore, (i) ~v · ~w = 7.5
For the second calculation, we want to find the magnitude of the vector that results from twice vector ~v minus vector ~w. This is calculated using the formula:
||2~v - ~w|| = sqrt((2*v)^2 + (-w)^2)
Substituting in the given magnitudes for ~v and ~w, we get:
||2~v - ~w|| = sqrt((2*3)^2 + (-5)^2) = sqrt((2*3)^2 + (-5)^2) ~= 7.81
Therefore, (ii) ||2v - w|| ~= 7.81
From the provided choices, even though there is no exact match, the closest choice is (b) (i) 7.5, (ii) 10, making this the best answer from the provided options.