MATH SOLVE

8 months ago

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

Accepted Solution

A:

Dot product has the following definition:

[tex]\textbf{a}\cdot \textbf{b}=|a| |b| cos(\theta)[/tex]

Where [tex] \theta[/tex] is the angle between two vectors and [tex] |a|, |b| [/tex] are their lengths.

The dot product of the vector with itself will give you the square of its length.

[tex]\textbf{a}\cdot \textbf{a}=|a| |a| cos(0)=|a| |a|=|a|^2[/tex]

If you are given end points of a vector dot product is defined in the following fashion:

[tex]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[/tex]

[tex] \textbf{a}\cdot \textbf{a}=\sum a_i^2[/tex]

Where [tex] \textbf{e}_i [/tex] 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:

[tex]v= [7,24]\\ \textbf{v}\cdot \textbf{v}=|v|^2=(7\cdot 7)+(24\cdot 24)=625\\ |v|=\sqrt{625}=25[/tex]

So the answer is A.

This formula will give you the length of a vector in Euclidian geometry:

[tex]|a|=\sqrt{a_1^2+a_2^2+a_3^3[/tex]

Where [tex] a_i [/tex] are the coordinates of the end point of that vector.

[tex]\textbf{a}\cdot \textbf{b}=|a| |b| cos(\theta)[/tex]

Where [tex] \theta[/tex] is the angle between two vectors and [tex] |a|, |b| [/tex] are their lengths.

The dot product of the vector with itself will give you the square of its length.

[tex]\textbf{a}\cdot \textbf{a}=|a| |a| cos(0)=|a| |a|=|a|^2[/tex]

If you are given end points of a vector dot product is defined in the following fashion:

[tex]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[/tex]

[tex] \textbf{a}\cdot \textbf{a}=\sum a_i^2[/tex]

Where [tex] \textbf{e}_i [/tex] 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:

[tex]v= [7,24]\\ \textbf{v}\cdot \textbf{v}=|v|^2=(7\cdot 7)+(24\cdot 24)=625\\ |v|=\sqrt{625}=25[/tex]

So the answer is A.

This formula will give you the length of a vector in Euclidian geometry:

[tex]|a|=\sqrt{a_1^2+a_2^2+a_3^3[/tex]

Where [tex] a_i [/tex] are the coordinates of the end point of that vector.