Given the vectors $ u = < 2,2 > ,v = < - 3,4 > $ and $ w = < 1, - 2 > $ , how do you find $ (u \cdot v) - (u \cdot w)? $

Hint: As we know that vectors are the mathematical representation of physical quantities for which both magnitude and direction can be determined. It is represented with a straight line with an arrow head. It is symbolically represented as $ \overrightarrow {AB} $ or $ \overrightarrow a $ . To solve this question we will use the dot product of vectors.

Complete step-by-step answer:
We know that $ \underline a = < {x_1},{y_1} > $ and $ \underline b = < {x_2},{y_2} > $ , then the dot or scalar product can be written as $ \underline a \cdot \underline b = {x_1}{x_2} + {y_1}{y_2} $ .
Now by putting the values we have: $ \left( {\underline u \cdot \underline v } \right) - \left( {\underline u \cdot \underline w } \right) = \left( {2 \times - 3} \right) + \left( {2 \times 4} \right) - \left[ {\left( {2 \times 1} \right) + \left( {2 \times - 2} \right)} \right] $ .
On further solving we have $ - 6 + 8 - \left( { - 2} \right) = 4 $ .
Hence the required value of $ (u \cdot v) - (u \cdot w) $ is $ 4 $
So, the correct answer is “4”.

Note: Before solving this kind of questions we should have the proper knowledge of vectors and their properties, formulas. Here we have used the dot product or the cross product to solve this i.e. multiplication of any two vectors is done by finding their cross products. It can be done by $ \vec v \cdot \vec w = \left| {\vec v} \right| \cdot \left| {\vec w} \right| \cdot \cos \theta $ .