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 $\pmatrix{a\cr b\cr c}$