Certainly! Let's find the dot product of the given vectors [tex]\( v_1=(2,4) \)[/tex] and [tex]\( v_2=(-1,5) \)[/tex].
The dot product [tex]\( v_1 \cdot v_2 \)[/tex] is calculated using the formula:
[tex]\[ v_1 \cdot v_2 = v_{1x} \cdot v_{2x} + v_{1y} \cdot v_{2y} \][/tex]
Where [tex]\( v_{1x} \)[/tex] and [tex]\( v_{1y} \)[/tex] are the components of vector [tex]\( v_1 \)[/tex], and [tex]\( v_{2x} \)[/tex] and [tex]\( v_{2y} \)[/tex] are the components of vector [tex]\( v_2 \)[/tex].
For our vectors:
[tex]\[ v_1 = (2, 4) \][/tex]
[tex]\[ v_2 = (-1, 5) \][/tex]
Substitute the components into the formula:
[tex]\[ v_1 \cdot v_2 = (2) \cdot (-1) + (4) \cdot (5) \][/tex]
Now, perform the multiplication:
[tex]\[ (2) \cdot (-1) = -2 \][/tex]
[tex]\[ (4) \cdot (5) = 20 \][/tex]
Add these two results together:
[tex]\[ -2 + 20 = 18 \][/tex]
Therefore, the dot product [tex]\( v_1 \cdot v_2 \)[/tex] is equal to 18. The correct answer is:
[tex]\[ 18 \][/tex]
