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If one of the two components of a vector is not zero, can the magnitude of the other vector component of this vector be zero?

a) Yes, it is always zero.
b) No, it is always nonzero.

1 Answer

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Final answer:

The magnitude of the other vector component can be zero if one component of a vector is not zero. When vectors are orthogonal, their components along the direction of each other are zero. A vector's magnitude cannot be exceeded by its own components.

Step-by-step explanation:

If one of the two components of a vector is not zero, the magnitude of the other vector component can indeed be zero. This question relates to the fundamental concept of vector components in two-dimensional space, where each vector is composed of two parts: one in the x-direction and one in the y-direction. For example, a vector with a non-zero x-component and a zero y-component would lie entirely along the x-axis, thereby having a zero magnitude in the y-direction.

Now, when vectors ᴀ and B are orthogonal, the component of B along the direction of ᴀ is zero, because orthogonal vectors are at a 90-degree angle to one another. Similarly, the component of ᴀ along the direction of B is also zero. It's important to understand that orthogonality implies that the dot product of the two vectors is zero.

Lastly, it's crucial to note that the magnitude of a vector cannot be exceeded by any of its individual components, and if two vectors are equal, their respective components must also be equal. A null vector, which has zero magnitude and no direction, is an example of a special case where all vector components are zero.

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