Question

question
type the correct answer in each box. round the vector’s magnitude to the nearest hundredth.
vector w has its initial point at (2, 5) and its terminal point at (-4, -2). write the vector in component
w = < , >, and ||w|| ≈ units.

Explanation:

Step1: Find component form

To find the component form of a vector with initial point \((x_1, y_1)\) and terminal point \((x_2, y_2)\), we use the formula \(\vec{w}=\langle x_2 - x_1, y_2 - y_1
angle\). Here, \(x_1 = 2\), \(y_1 = 5\), \(x_2=-4\), \(y_2 = - 2\). So \(x_2 - x_1=-4 - 2=-6\) and \(y_2 - y_1=-2 - 5=-7\). Thus, \(\vec{w}=\langle - 6,-7
angle\).

Step2: Find magnitude

The magnitude of a vector \(\langle a,b
angle\) is given by \(\|\vec{w}\|=\sqrt{a^{2}+b^{2}}\). Substituting \(a = - 6\) and \(b=-7\), we get \(\|\vec{w}\|=\sqrt{(-6)^{2}+(-7)^{2}}=\sqrt{36 + 49}=\sqrt{85}\approx9.22\).

Answer:

\(\vec{w}=\langle - 6,-7
angle\), and \(\|\vec{w}\|\approx9.22\) units.