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Question 6: Suppose you are given a known non-zero vector A. The scalar product of A with an unknown vector B is zero. Likewise, the vector product of A with B is zero. What can you conclude about B?

Solution

The scalar product of A and B is given by,

Now it is given that this product is zero. It is possible only when any of the following conditions are true.

  1. A is zero. (This is not the case as given in the question).
  2. B is zero.
  3. Cosθ is zero ⇒ θ = 90°.

So, for the scalar product of the given vectors to be zero, we have either B is zero or both the vectors are perpendicular.

The vector product of the two vectors is given by,

Now it is given that this product is also zero. However, it is possible only when any of the following conditions are true.

  1. A is zero. (This is not the case as given in the question).
  2. B is zero.
  3. Sinθ is zero ⇒ θ = 0°.

So, for the vector product of the given vectors to be zero, we have either B is zero or both vectors are parallel.

Now to check the possibility of both conditions of scalar and vector product to be zero, we see both vectors cannot be parallel and perpendicular at the same time. So we have the only acceptable condition that B = 0.

Thus we conclude about the vector B that it is a null vector.

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