To find the length of a vector given its components, we employ the formula for the Euclidean norm (magnitude) of the vector. The vector in question has components [tex]\( x = -6 \)[/tex] and [tex]\( y = 9 \)[/tex].
The formula for the length (or magnitude) of the vector [tex]\( \mathbf{v} = \binom{x}{y} \)[/tex] is given by:
[tex]\[
\|\mathbf{v}\| = \sqrt{x^2 + y^2}
\][/tex]
Here is the step-by-step process to find the length:
1. Calculate the square of each component:
[tex]\[
x^2 = (-6)^2 = 36
\][/tex]
[tex]\[
y^2 = 9^2 = 81
\][/tex]
2. Add the squares of the components:
[tex]\[
x^2 + y^2 = 36 + 81 = 117
\][/tex]
3. Take the square root of the sum:
[tex]\[
\|\mathbf{v}\| = \sqrt{117} \approx 10.816653826391969
\][/tex]
So, the length of the vector [tex]\( \binom{-6}{9} \)[/tex] is approximately [tex]\( 10.816653826391969 \)[/tex].
