Given [tex]$u=15\left(\cos 45^{\circ}, \sin 45^{\circ}\right)$[/tex], what are the magnitude and direction of [tex]-5 u[/tex]?

A. [tex]75 ; 225^{\circ}[/tex]
B. [tex]75 ; 135^{\circ}[/tex]
C. [tex]-75 ;-135^{\circ}[/tex]
D. [tex]-75 ; 45^{\circ}[/tex]



Answer :

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]