Question

Is it possible for the product of two non-zero vectors to be zero?

Answer 1

Yes, if you are referring to dot product or to cross product.

Explanation:

The dot product of any two orthogonal vectors is `0`.

The cross product of any two collinear vectors is `0` or a zero length vector (according to whether you are dealing with `2` or `3` dimensions).

Note that for any two non-zero vectors, the dot product and cross product cannot both be zero.

There is a vector context in which the product of any two non-zero vectors is non-zero. It is known as Hamilton's Quaternions. Quaternions form a `4` dimensional vector space over the real numbers and their multiplication is a combination of dot product and cross product. Actually in the history of mathematics, quaternion multiplication predates dot product and cross product - it is where they came from.

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Answer 2

It is possible to get a zero-magnitude resultant vector for both dot-product and cross-product vector multiplication.

Explanation:

Vector multiplication is usually defined as being one of either the dot-product (sometimes called the inner or scalar product) or a cross-product (sometimes called the outer or vector product). In each case it is possible to get a zero-magnitude result.

Dot Product:

The dot-product of two vectors `\vec a` and `\vec b` is defined as

`\vec a * \vec b = |\vec a| |\vec b|cos \theta`

where `\theta` is the angle between the vectors (measured the smallest possible way in the plane defined by the two vectors). It can be seen that the only way for the result to be zero for non-zero vectors is for

`cos \theta =0`

which means that the result is zero if the vectors are perpendicular to each other

`=> \theta = \pi/2` in radians
`=> \theta = 90^"o"` in degrees

Cross Product:

the cross-product of two vectors `\vec a` and `\vec b` is defined as

`\vec a \times \vec b = |\vec a| |\vec b| sin (\theta) \hat n`

where the result is a vector pointing in the direction `\hat n` which is perpendicular to the plane defined by the two vectors. Again, we can see that there is only one way for non-zero vectors to produce a zero magnitude result, which is for

`sin \theta =0`

which means that the result is zero if the vectors are parallel to each other

`=> \theta = 0` in radians
`=> \theta = 0^"o"` in degrees