### Video Transcript

A body is moving in a straight line
from point ๐ด negative six, zero to point ๐ต negative five, four under the action of
the vector force ๐
, which is equal to ๐๐ข plus two ๐ฃ newtons. Given that the change in the bodyโs
potential energy is two joules and that the displacement is in meters, determine the
value of the constant ๐.

We are told that the body moves in
a straight line from point ๐ด to point ๐ต, where ๐ด and ๐ต have coordinates negative
six, zero and negative five, four. This means that we move one unit to
the right and four units up. If we consider the unit vectors ๐ข
and ๐ฃ in the horizontal and vertical direction, respectively, our displacement
vector is equal to ๐ข plus four ๐ฃ.

We are also told that the vector
force acting on the body is ๐๐ข plus two ๐ฃ newtons. We know that the work done is the
dot or scalar product of the force vector and the displacement vector. The work done is therefore equal to
the dot product of ๐ข plus four ๐ฃ and ๐๐ข plus two ๐ฃ.

To calculate the dot product, we
find the sum of the products of the individual components. In this question, this is equal to
one multiplied by ๐ plus four multiplied by two. The ๐ข-components are one and ๐,
and the ๐ฃ-components are four and two. This simplifies to ๐ plus
eight.

We are also told in the question
that the change in potential energy is equal to two joules. As energy can only be transferred
and not destroyed or created, we know that the sum of the work done and the
gravitational potential energy is equal to zero. This means that ๐ plus eight plus
two must equal zero. Collecting like terms, we have ๐
plus 10 is equal to zero. Finally, we can subtract 10 from
both sides of this equation, giving us a value of ๐ equal to negative 10. This means that the vector force ๐
is equal to negative 10๐ข plus two ๐ฃ.