To solve this problem, we need to find the magnitude (or norm) of the scaled vector [tex]\( c \vec{u} \)[/tex], where [tex]\( \vec{u} = \langle 5, -12 \rangle \)[/tex] and [tex]\( c = -3 \)[/tex].
Here are the steps to solve this problem:
1. Scale the Vector:
First, multiply each component of the vector [tex]\( \vec{u} \)[/tex] by the scalar [tex]\( c \)[/tex]:
[tex]\[
c \vec{u} = (-3) \times \langle 5, -12 \rangle = \langle (-3) \times 5, (-3) \times (-12) \rangle
\][/tex]
Simplifying the multiplication:
[tex]\[
c \vec{u} = \langle -15, 36 \rangle
\][/tex]
2. Calculate the Magnitude:
The magnitude (or norm) of a vector [tex]\( \langle a, b \rangle \)[/tex] is calculated using the formula:
[tex]\[
\| \langle a, b \rangle \| = \sqrt{a^2 + b^2}
\][/tex]
Applying this to our scaled vector [tex]\( \langle -15, 36 \rangle \)[/tex]:
[tex]\[
\| \langle -15, 36 \rangle \| = \sqrt{(-15)^2 + 36^2}
\][/tex]
Calculate the squares:
[tex]\[
(-15)^2 = 225 \quad \text{and} \quad 36^2 = 1296
\][/tex]
Sum the squares:
[tex]\[
225 + 1296 = 1521
\][/tex]
Finally, take the square root of the sum:
[tex]\[
\sqrt{1521} = 39
\][/tex]
Therefore, the magnitude [tex]\( \| c \vec{u} \| \)[/tex] is [tex]\( 39 \)[/tex].
The correct answer is:
[tex]\[
39
\][/tex]
