# Trouble w/ Vector Dot Product

## Homework Statement

A cutting tool under microprocessor control has several forces acting on it. One force is $\vec{F}$=-αxy2$\hat{j}$, a force in the negative y-direction whose magnitude depends on the position of the tool. The constant is α=2.50 N/m3. Consider the displacement of the tool from the origin to the point x=3.00m, y=3.00m.

(a) Calculate the work done on the tool by $\vec{F}$ if this displacement is along the straight line y=x that connects these two points.

## Homework Equations

W=∫$\vec{F}$$\cdot$d$\vec{l}$

## The Attempt at a Solution

I'm trying to use the equation above, so here's what I know:

d$\vec{l}$=dx$\hat{i}$+dy$\hat{j}$
$\vec{F}$=-αxy2$\hat{j}$

Since it's the dot product,
$\vec{F}$$\cdot$d$\vec{l}$=dx+-αxy2dy.

I'm confused as to why the right side of that equation is equal to -αy3dy, as the textbook solution suggests. Any help is appreciated.

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Andrew Mason
Homework Helper
I'm confused as to why the right side of that equation is equal to -αy3dy, as the textbook solution suggests. Any help is appreciated.
If x = y then $-axy^2 = -ay^3$

The dot product of the two vectors is:

$\vec{a} \cdot \vec{b} = ab\cos\theta$

where $\theta$ is the angle between the two vectors.

AM