A vector has an initial point at $(2,1)$ and a terminal point at $(7,5)$. What is the component form and magnitude of this vector?

• A. Component form: $(5,4)$; Magnitude: $\sqrt{5^2+4^2}$ = $6.4$
• B. Component form: $(7,5)$; Magnitude: $\sqrt{7^2+5^2}$ = $8.6$
• C. Component form: $(2,1)$; Magnitude: $\sqrt{2^2+1^2}$ = $2.2$
• D. Component form: $(5,4)$; Magnitude: $\sqrt{4^2+5^2}$ = $9$

Answer: (A)

Verified step by step guidance

1. Identify the coordinates of the initial point and the terminal point of the vector. Here, the initial point is (2, 1) and the terminal point is (7, 5).
2. Calculate the component form of the vector by subtracting the coordinates of the initial point from the terminal point. Use the formula: \(\textbf{v} = (x_2 - x_1, y_2 - y_1)\), where \((x_1, y_1)\) is the initial point and \((x_2, y_2)\) is the terminal point.
3. Substitute the given points into the formula to find the components: \(x\)-component = \$7 - 2\(, \)y\(-component = \)5 - 1$.
4. Write the component form of the vector as \(\textbf{v} = (5, 4)\) based on the calculations from the previous step.
5. Find the magnitude of the vector using the Pythagorean theorem: \(|\textbf{v}| = \sqrt{(x\text{-component})^2 + (y\text{-component})^2}\). Substitute the components to get \(|\textbf{v}| = \sqrt{5^2 + 4^2}\).