Question

Use the dot product to find |v| when v=(7,24) a. 25 b. -17 c. 31 d. 625

Answer 1

Dot product has the following definition:
$\textbf{a}\cdot \textbf{b}=|a| |b| cos(\theta)$
Where $\theta$ is the angle between two vectors and $|a|, |b|$ are their lengths. 
The dot product of the vector with itself will give you the square of its length.
$\textbf{a}\cdot \textbf{a}=|a| |a| cos(0)=|a| |a|=|a|^2$
If you are given end points of a vector dot product is defined in the following fashion:
$a=[a_1,a_2,a_3]\\ b=[b_1,b_2,b_3]\\ \textbf{a}\cdot \textbf{b}=\textbf{a} \sum b_i \textbf{e}_i= \sum b_i (\textbf{a} \cdot \textbf{e}_i)= \sum b_i a_i$
$\textbf{a}\cdot \textbf{a}=\sum a_i^2$
Where $\textbf{e}_i$ are unit vectors. These are vectors of a unit length and they span in direction of a coordinate axis (if you are working with Cartesian coordinate system). If you do a dot product of a unit vector of x-axis and unit vector of y-axis you get zero, because the angle between them is 90 degrees. 
Now we can apply the above formula to this problem:
$v= [7,24]\\ \textbf{v}\cdot \textbf{v}=|v|^2=(7\cdot 7)+(24\cdot 24)=625\\ |v|=\sqrt{625}=25$
So the answer is A.
This formula will give you the length of a vector in Euclidian geometry:
$|a|=\sqrt{a_1^2+a_2^2+a_3^3$
Where $a_i$ are the coordinates of the end point of that vector.

Answer 2

THE ANSWER IS A

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