Question

The force on an object is F⃗ =−17j⃗ . For the vector v⃗ =2i⃗ +3j⃗ , find: (a) The component of F⃗ parallel to v⃗

Answer 1

Answer:

(a) $\vec F_{\parallel} = -\frac{102}{13}\,i-\frac{103}{13}\,j$ , (b) $\vec F_{\perp} = \frac{102}{13}\,i -\frac{68}{13}\,j$, (c) $W = -51$

Explanation:

The statement is incomplete:

The force on an object is $\vec F = -17\,j$. For the vector $\vec v = 2\,i +3\,j$. Find: (a) The component of $\vec F$ parallel to $\vec v$, (b) The component of $\vec F$ perpendicular to $\vec v$, and (c) The work $W$, done by force $\vec F$ through displacement $\vec v$.

(a) The component of $\vec F$ parallel to $\vec v$ is determined by the following expression:

$\vec F_{\parallel} = (\vec F \bullet \hat {v} )\cdot \hat{v}$

Where $\hat{v}$ is the unit vector of $\vec v$, which is determined by the following expression:

$\hat{v} = \frac{\vec v}{\|\vec v \|}$

Where $\|\vec v\|$ is the norm of $\vec v$, whose value can be found by Pythagorean Theorem.

Then, if $\vec F = -17\,j$ and $\vec v = 2\,i +3\,j$, then:

$\|\vec v\| =\sqrt{2^{2}+3^{3}}$

$\|\vec v\|=\sqrt{13}$

$\hat{v} = \frac{1}{\sqrt{13}} \cdot(2\,i + 3\,j)$

$\hat{v} = \frac{2}{\sqrt{13}}\,i+ \frac{3}{\sqrt{13}}\,j$

$\vec F \bullet \hat{v} = (0)\cdot \left(\frac{2}{\sqrt{13}} \right)+(-17)\cdot \left(\frac{3}{\sqrt{13}} \right)$

$\vec F \bullet \hat{v} = -\frac{51}{\sqrt{13}}$

$\vec F_{\parallel} = \left(-\frac{51}{\sqrt{13}} \right)\cdot \left(\frac{2}{\sqrt{13}}\,i+\frac{3}{\sqrt{13}}\,j \right)$

$\vec F_{\parallel} = -\frac{102}{13}\,i-\frac{153}{13}\,j$

(b) Parallel and perpendicular components are orthogonal to each other and the perpendicular component can be found by using the following vectorial subtraction:

$\vec F_{\perp} = \vec F - \vec F_{\parallel}$

Given that $\vec F = -17\,j$ and $\vec F_{\parallel} = -\frac{102}{13}\,i-\frac{153}{13}\,j$, the component of $\vec F$ perpendicular to $\vec v$ is:

$\vec F_{\perp} = -17\,j -\left(-\frac{102}{13}\,i-\frac{153}{13}\,j \right)$

$\vec F_{\perp} = \frac{102}{13}\,i + \left(\frac{153}{13}-17 \right)\,j$

$\vec F_{\perp} = \frac{102}{13}\,i -\frac{68}{13}\,j$

(c) The work done by  $\vec F$ through displacement $\vec v$ is:

$W = \vec F \bullet \vec v$

$W = (0)\cdot (2)+(-17)\cdot (3)$

$W = -51$