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mathsVector AlgebraProperties 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.

click on the content to continue..

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'.

slide-show version coming soon