Question

Utilize the properties of the dot product to evaluate the given expression, assuming u⋅v=2, ∣∣u∣∣=1, and ∣∣v∣∣=3. 1. 2u⋅(3u−v) 2. ((u⋅v)⋅(u−v) Provide the answers for the expressions in terms of the given values.

Answer 1

To solve the given expressions, we substitute the values of u⋅v, ∣∣u∣∣, and ∣∣v∣∣ into the expressions and use the properties of the dot product. The first expression simplifies to 2, and the second expression simplifies to 2u−2v.

Step-by-step explanation:

To evaluate the given expressions, we will use the properties of the dot product. Let's start with the first expression: 2u⋅(3u−v).

Since we know that u⋅v = 2, we can substitute this value into the expression: 2u⋅(3u−v) = 2(3u⋅u−u⋅v).

Using the property that u⋅u = |u|^2 and the given information that |u| = 1, we can simplify further: 2(3⋅1^2−2) = 2(3−2) = 2(1) = 2.

Now let's move on to the second expression: ((u⋅v)⋅(u−v)). Again, substituting the given value of u⋅v = 2, we get ((2)⋅(u−v)).

Since the dot product is distributive, we can expand this expression to 2u−2v.