Video Transcript
True or False: If the component of a vector in the direction of another vector is zero, then the two are perpendicular.
We begin by recalling that we can find the component of some vector 𝐀 in the direction of another vector 𝐁 using scalar projection. The diagram drawn shows vector 𝐀 projected onto vector 𝐁. And using our knowledge of right angle trigonometry, we see that the component of vector 𝐀 in the direction of vector 𝐁 is equal to the magnitude of vector 𝐀 multiplied by cos 𝜃. More formally, we can say that the scalar projection of vector 𝐀 onto vector 𝐁 is equal to the magnitude of vector 𝐀 multiplied by cos 𝜃.
In this question, we are told that this is equal to zero. And since vector 𝐀 is not the zero vector, the magnitude of vector 𝐀 cannot equal zero. We can therefore conclude that cos 𝜃 equals zero. Using our knowledge of special angles or taking the inverse cosine of both sides, we see that 𝜃 is equal to 90 degrees. If the component of a vector in the direction of another vector is zero, the angle between the two vectors must be 90 degrees. This means that the two vectors are perpendicular, and the correct answer is true.
