To determine [tex]\(\|c u \|\)[/tex] for [tex]\(u = \langle 5, -12 \rangle\)[/tex] and [tex]\(c = -3\)[/tex], follow these steps:
1. Scalar Multiplication: Multiply each component of the vector [tex]\(u\)[/tex] by the scalar [tex]\(c\)[/tex]:
[tex]\[
c \cdot u = c \cdot \langle 5, -12 \rangle = \langle c \cdot 5, c \cdot -12 \rangle = \langle -15, 36 \rangle
\][/tex]
2. Calculate the Magnitude: The magnitude (or Euclidean norm) of a vector [tex]\( \langle a, b \rangle \)[/tex] is given by:
[tex]\[
\| \langle a, b \rangle \| = \sqrt{a^2 + b^2}
\][/tex]
For our vector [tex]\(\langle -15, 36 \rangle\)[/tex]:
[tex]\[
\| \langle -15, 36 \rangle \| = \sqrt{(-15)^2 + 36^2} = \sqrt{225 + 1296} = \sqrt{1521} = 39
\][/tex]
Therefore, the magnitude [tex]\(\|c u \|\)[/tex] is [tex]\(39\)[/tex].
Among the given options, the correct answer is:
- [tex]\(39\)[/tex]
So, [tex]\(\|c u \| = 39\)[/tex].
