__maths__>__Properties of Vector Arithmetics__>__Properties of Cross Product__### Cross product of a Scalar Multiple

In this page, you will learn about the vector cross product with scalar multiple of a vector.

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Given `vec p = 2i+j-k` and `lambda = 3.1`, what is the scalar multiple `3.1 vec p`?

- `6.2i+3.1j-3.1k`
- `6.2i+3.1j-3.1k`
- `6.2i+j-k`

The answer is '`6.2i+3.1j-3.1k`'

Given the definition of cross product as

`vec p xx vec q = |vec p||vec q|sin theta hat n`

What is `vec p xx (lambda vec q)`?

- `lambda(vec p xx vec q)`
- `|p||lambda q|sin(theta) hat n` if `lambda` is positive
- `lambda|p||q|sin(theta) hat n`
- all the above
- all the above

The answer is 'All the above'

Note that if `lambda` is a negative number, it equals `|p||lambda q|sin(180+theta) hat n`.

Given the definition of cross product as

`vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|`

What is `vec p xx (lambda vec q)`?

- `lambda (vec p xx vec q)`
- `|(i, j, k),(p_x, p_y, p_z),(lambda q_x, lambda q_y, lambda q_z)|`
- `lambda |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|`
- all the above
- all the above

The answer is 'All the above'

• Cross product with a scalar multiple of a vector equals scalar multiple of the cross product with the vector.

**Cross Product of scalar multiple: ** For any vectors `vec p, vec q`

`vec p xx (lambda vec q) = lambda (vec p xx vec q)`

*Solved Exercise Problem: *

Given that `vec p xx vec q = 3j+1.1k`, what is `vec p/(2.1) xx vec q`?

- `3/(2.1)j+1.1/(2.1)k`
- `3/(2.1)j+1.1/(2.1)k`
- `6.3j+2.31k`
- `3/(2.1)j+1.1k`

The answer is '`3/(2.1)j+1.1/(2.1)k`'.

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