Video Transcript
Determine whether the following is
true or false: If the component of a vector in the direction of another vector is
zero, then the two are parallel.
Let’s begin by considering two
vectors 𝐀 and 𝐁 as shown. The component of vector 𝐀 on
vector 𝐁 is shown in the diagram. We can see that this creates a
right angle triangle. And we can let the angle at the
origin be 𝜃. In right angle trigonometry, the
cosine ratio is equal to the adjacent over the hypotenuse. We know that the hypotenuse is
equal to the magnitude of vector 𝐀. And if we let the component of
vector 𝐀 on vector 𝐁 be equal to 𝑥, then cos 𝜃 is equal to 𝑥 over the magnitude
of vector 𝐀. Rearranging this equation, we get
that 𝑥 is equal to the magnitude of vector 𝐀 multiplied by cos 𝜃.
We’re told in the question that 𝑥
is equal to zero. Therefore, zero is equal to the
magnitude of vector 𝐀 multiplied by cos 𝜃. The magnitude of any of vector must
be positive; it could not equal zero. This means that cos 𝜃 must be
equal to zero. 𝜃 is therefore equal to 90
degrees. This means that the two vectors are
actually perpendicular. We can therefore conclude that the
statement is false. The vectors are not parallel but
are perpendicular.
