1 Answer

Wally · Answer 1 · 2021-10-25T14:28:09+0000

Answer:

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

Step-by-step 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
, 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:

$\hat{v} = (\vec v)/(\|\vec v \|)$

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:

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

$\|\vec v\|=√(13)$

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

$\hat{v} = (2)/(√(13))\,i+ (3)/(√(13))\,j$

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

$\vec F \bullet \hat{v} = -(51)/(√(13))$

$\vec F_(\parallel) = \left(-(51)/(√(13)) \right)\cdot \left((2)/(√(13))\,i+(3)/(√(13))\,j \right)$

$\vec F_(\parallel) = -(102)/(13)\,i-(153)/(13)\,j$

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

$\vec F_(\perp) = \vec F - \vec F_(\parallel)$