Question

If a vector has an x-component of -25 units and a y-component of 40 units, then the magnitude and direction of this vector are:

• A. $5\sqrt{89}$ units; ${\sin }^{-1}\frac{-5}{\sqrt{89}}$ with x-axis
• B. $5\sqrt{89}$ units; ${\cos }^{-1}\frac{-5}{\sqrt{89}}$ with x-axis
• C. 45 units; ${\cos }^{-1}\frac{-5}{9}$ with x-axis
• D. 45 units; ${\sin }^{-1}\frac{-5}{9}$ with x-axis

Answer 1

Answer: (B)

$5\sqrt{89}$ units; ${\cos }^{-1}\frac{-5}{\sqrt{89}}$ with x-axis

Magnitude $=\sqrt{x^2+y^2}=\sqrt{(-25{)}^2+{40}^2}=\sqrt{2225}=5\sqrt{89}$
Angle with x-axis $={\tan }^{-1}\left(y/x\right)$

$={\tan }^{-1}(40/-25)={\tan }^{-1}(-8/5)$
$={\cos }^{-1}(-5/\sqrt{5^2+6^2})={\cos }^{-1}(-5/\sqrt{89})$