Let's work through the problem step by step:
1. Find the components of vector [tex]\( u \)[/tex]:
- Initial point of [tex]\( u \)[/tex]: (17, 5)
- Terminal point of [tex]\( u \)[/tex]: (9, -12)
- To find the components, [tex]\( u = (u_x, u_y) = (\text{terminal}_x - \text{initial}_x, \text{terminal}_y - \text{initial}_y) \)[/tex]
- Calculating:
[tex]\[
u_x = 9 - 17 = -8
\][/tex]
[tex]\[
u_y = -12 - 5 = -17
\][/tex]
- So, vector [tex]\( u \)[/tex] has components [tex]\( (-8, -17) \)[/tex].
2. Find the components of vector [tex]\( v \)[/tex]:
- Initial point of [tex]\( v \)[/tex]: (12, 4)
- Terminal point of [tex]\( v \)[/tex]: (3, -2)
- To find the components, [tex]\( v = (v_x, v_y) = (\text{terminal}_x - \text{initial}_x, \text{terminal}_y - \text{initial}_y) \)[/tex]
- Calculating:
[tex]\[
v_x = 3 - 12 = -9
\][/tex]
[tex]\[
v_y = -2 - 4 = -6
\][/tex]
- So, vector [tex]\( v \)[/tex] has components [tex]\( (-9, -6) \)[/tex].
3. Find [tex]\( 3u \)[/tex]:
- Multiply each component of [tex]\( u \)[/tex] by 3:
[tex]\[
3u_x = 3 \times (-8) = -24
\][/tex]
[tex]\[
3u_y = 3 \times (-17) = -51
\][/tex]
- So, [tex]\( 3u \)[/tex] has components [tex]\( (-24, -51) \)[/tex].
4. Find [tex]\( -2v \)[/tex]:
- Multiply each component of [tex]\( v \)[/tex] by -2:
[tex]\[
(-2)v_x = -2 \times (-9) = 18
\][/tex]
[tex]\[
(-2)v_y = -2 \times (-6) = 12
\][/tex]
- So, [tex]\( -2v \)[/tex] has components [tex]\( (18, 12) \)[/tex].
5. Add the components to find [tex]\( 3u - 2v \)[/tex]:
- Calculate the sum of components:
[tex]\[
(\text{component wise sum}) = (3u_x + (-2)v_x, 3u_y + (-2)v_y)
\][/tex]
[tex]\[
= (-24 + 18, -51 + 12)
\][/tex]
[tex]\[
= (-6, -39)
\][/tex]
- So, [tex]\( 3u - 2v \)[/tex] has components [tex]\( (-6, -39) \)[/tex].
6. Calculate the magnitude of [tex]\( 3u - 2v \)[/tex]:
- Use the formula for the magnitude of a vector [tex]\( (x, y) \)[/tex] where [tex]\( \| \mathbf{v} \| = \sqrt{x^2 + y^2} \)[/tex]:
[tex]\[
\| 3u - 2v \| = \sqrt{(-6)^2 + (-39)^2}
\][/tex]
[tex]\[
= \sqrt{36 + 1521}
\][/tex]
[tex]\[
= \sqrt{1557}
\][/tex]
[tex]\[
\approx 39.46
\][/tex]
7. Round the answer to the nearest hundredth:
- The magnitude calculated is approximately 39.46.
Therefore, the correct answer is:
[tex]\[ \boxed{39.46} \][/tex]
Which corresponds to option C: [tex]\( 39.46 \)[/tex] units.
