Question

fill in the blank to correctly complete the following sentence.
the formula for the dot product of the two vectors ( u = langle a,b
angle ) and ( v = langle c,d
angle ) is ( ucdot v=)____.
the formula for the dot product of the two vectors ( u = langle a,b
angle ) and ( v = langle c,d
angle ) is ( ucdot v=square ).

Explanation:

Step1: Recall the dot product formula for two - dimensional vectors

For two vectors \(\mathbf{u}=\langle a,b
angle\) and \(\mathbf{v}=\langle c,d
angle\), the dot product is calculated by multiplying the corresponding components and then adding the results.

Step2: Calculate the dot product

The first components are \(a\) (from \(\mathbf{u}\)) and \(c\) (from \(\mathbf{v}\)), and their product is \(a\times c = ac\). The second components are \(b\) (from \(\mathbf{u}\)) and \(d\) (from \(\mathbf{v}\)), and their product is \(b\times d=bd\). Then we add these two products together. So \(\mathbf{u}\cdot\mathbf{v}=ac + bd\).

Answer:

\(ac + bd\)