Question

vector u has its initial point at (17, 5) and its terminal point at (9, -12). vector v has its initial point at (12, 4) and its terminal point at (3, -2). find ||3u - 2v||. round your answer to the nearest hundredth.
a. 29.79 units
b. 32.12 units
c. 39.46 units
d. 42.23 units

Explanation:

Step1: Find components of vector u

The component form of a vector with initial point \((x_1, y_1)\) and terminal point \((x_2, y_2)\) is \(\langle x_2 - x_1, y_2 - y_1
angle\). For vector \(u\), initial point \((17, 5)\) and terminal point \((9, -12)\), so \(u=\langle 9 - 17, -12 - 5
angle=\langle - 8, -17
angle\).

Step2: Find components of vector v

For vector \(v\), initial point \((12, 4)\) and terminal point \((3, -2)\), so \(v=\langle 3 - 12, -2 - 4
angle=\langle - 9, -6
angle\).

Step3: Calculate 3u - 2v

First, find \(3u\) and \(2v\). \(3u = 3\langle - 8, -17
angle=\langle - 24, -51
angle\), \(2v=2\langle - 9, -6
angle=\langle - 18, -12
angle\). Then \(3u - 2v=\langle - 24-(-18), -51 - (-12)
angle=\langle - 6, -39
angle\).

Step4: Find the magnitude of 3u - 2v

The magnitude of a vector \(\langle a, b
angle\) is \(\sqrt{a^{2}+b^{2}}\). So for \(3u - 2v=\langle - 6, -39
angle\), the magnitude is \(\sqrt{(-6)^{2}+(-39)^{2}}=\sqrt{36 + 1521}=\sqrt{1557}\approx39.46\).

Answer:

C. 39.46 units