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Question Video: Determining Parallel Vectors Mathematics

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.

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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 πœƒ. 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 π‘₯, then cos πœƒ is equal to π‘₯ over the magnitude of vector 𝐀. Rearranging this equation, we get that π‘₯ is equal to the magnitude of vector 𝐀 multiplied by cos πœƒ.

We’re told in the question that π‘₯ is equal to zero. Therefore, zero is equal to the magnitude of vector 𝐀 multiplied by cos πœƒ. The magnitude of any of vector must be positive; it could not equal zero. This means that cos πœƒ must be equal to zero. πœƒ 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.

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