### Video Transcript

The diagram shows a vector, 𝐀,
that has a magnitude of 55. The angle between the vector and
the 𝑥-axis is 82 degrees. Work out the horizontal component
of the vector. Give your answer to the nearest
whole number.

In this question, we are being
asked to solve for the horizontal component of the vector 𝐀. We can label this component 𝐴
subscript 𝑥. To find this horizontal component,
recall that we can use the formula 𝐴 subscript 𝑥 equals 𝐴 times cos 𝜃, where 𝐴
is the magnitude of the vector and 𝜃 is the argument of the vector.

We are told in the question that
the magnitude of the vector 𝐀 is 55. We are also told that there is an
angle of 82 degrees between the vector and the negative 𝑥-axis. But we need to be careful here;
this angle is not the argument of the vector. The argument of a vector is defined
as the angle between the vector and the positive 𝑥-axis, measured counterclockwise
from the positive 𝑥-axis.

So, to find the argument of this
vector, we need to add 180 degrees to this angle of 82 degrees. Adding 82 degrees plus 180 degrees
gives us that the argument 𝜃 is equal to 262 degrees. Substituting these values for the
magnitude 𝐴 and the argument 𝜃 into our equation, we find that the horizontal
component 𝐴 subscript 𝑥 is equal to 55 multiplied by the cos of 262 degrees.

Completing this calculation, we
find that the horizontal component 𝐴 subscript 𝑥 of the vector 𝐀 is equal to
negative 7.654. We want this answer to the nearest
whole number, so we can round down negative 7.654 to negative eight. And so we have arrived at the final
answer. To the nearest whole number, the
horizontal component of the vector 𝐀 is negative eight.