To determine the magnitude and direction of [tex]\(-5u\)[/tex] given [tex]\( u = 15\left( \cos 45^\circ, \sin 45^\circ \right) \)[/tex], we can follow these steps:
1. Calculate the Magnitude:
- The original magnitude of vector [tex]\( u \)[/tex] is given as [tex]\( 15 \)[/tex].
- To find the magnitude of [tex]\(-5u\)[/tex], we multiply the magnitude of [tex]\( u \)[/tex] by [tex]\( 5 \)[/tex].
[tex]\[
\text{Magnitude of } -5u = 15 \times 5 = 75
\][/tex]
2. Determine the Direction:
- The original direction of [tex]\( u \)[/tex] is [tex]\( 45^\circ \)[/tex].
- Multiplying [tex]\( u \)[/tex] by [tex]\(-5\)[/tex] changes its direction. In general, multiplying by a negative scalar reverses the direction of the vector, which means adding [tex]\( 180^\circ \)[/tex] to the original angle.
[tex]\[
\text{New direction} = 45^\circ + 180^\circ = 225^\circ
\][/tex]
Therefore, the magnitude of [tex]\(-5u\)[/tex] is [tex]\( 75 \)[/tex] and the direction is [tex]\( 225^\circ \)[/tex].
After comparing with the given options:
1. [tex]\( 75 ; 225^\circ \)[/tex]
2. [tex]\( 75 ; 135^\circ \)[/tex]
3. [tex]\( -75 ; -135^\circ \)[/tex]
4. [tex]\( -75 ; 45^\circ \)[/tex]
We can see that the correct answer is:
[tex]\[ \boxed{75 ; 225^\circ} \][/tex]
