# Cross with the Scalar Product

##### Age 16 to 18 Challenge Level:

### Why do this problem?

This problem helps to reinforce students' understanding of the
vector and scalar product by encouraging them to think about how
they are related geometrically.

### Possible approach

Perhaps begin with a recap of the scalar and vector product to
make sure students are clear about what each product is and how it
is calculated.

Then set the first challenge, to explore and describe
geometrically the set of vectors ${\bf u}$ with the property that
${\bf u}\cdot{\bf v}=0$. If students choose to consider the problem
algebraically, ask them to interpret their findings
geometrically.

Once students have reached their conclusions, introduce the
second part of the problem. Encourage students to consider how
their geometrical insights from the first part relate to what they
are being asked to do in the second part of the problem.
Identifying which of the four vectors could result from the cross
product of a vector with $\bf v$ should be straightforward and not
require lots of difficult algebra.

Finally, the problem asks students to find a method to quickly
construct other vectors which result from the cross product of
vectors with $\bf v$. Encourage students to think of how to
generate such vectors from vectors they have already found, using
the properties of the scalar product.

### Key questions

What do the vectors $\bf u$ such that $\bf u \cdot \bf v =0$
have in common?

How can you consider this set of vectors $\bf u$
geometrically?

What do the vectors $\bf w$ resulting from the cross product
of $\bf v$ with other vectors have in common?

### Possible extension

Find a method to quickly generate sets of vectors which are
perpendicular to $\pmatrix{a\cr b\cr c}$

### Possible support