NAME:

Quiz 3, Math 223, Section 2, 9-11-98

If $\vec{a}$ and $\vec{b}$ are any nonzero vectors in 3-space, which are not parallel to one another, write expressions for vectors representing the following:

1.

A unit vector parallel to $\vec{a}$.The vector $\lambda \vec{a}$ is parallel to $\vec{a}$. We want $\Vert \lambda \vec{a} \Vert = 1$ so $\Vert\lambda \vec{a}\Vert = \vert\lambda\vert \Vert\vec{a}\Vert = 1$ implies that $\lambda = 1/\Vert\vec{a}\Vert$. Thus, a unit vector parallel to $\vec{a}$ is $\vec{a}/\Vert\vec{a}\Vert$.

2.

A vector perpendicular to $\vec{a}$ and $\vec{b}$.

The vector $\vec{a} \times \vec{b}$ is defined to be the vector which is perpendicular to both $\vec{a}$ and $\vec{b}$ in the direction of the right hand rule. It has magnitude given by $\Vert \vec{a}\Vert \Vert\vec{b}\Vert \sin \theta$ if $\theta$ is the angle between the two vectors.

3.

What is the value of $\vec{a} \cdot (\vec{a} \times \vec{b})$? Why?

Now, $\vec{a} \cdot (\vec{a} \times \vec{b}) = \Vert \vec{a} \Vert \Vert\vec{b} \Vert \cos \theta$, but $\theta = \pi/2$ radians since the cross product of $\vec{a}$ and $\vec{b}$ is perpendicular to $\vec{a}$ by part 2 above. Thus, $\vec{a} \cdot (\vec{a} \times \vec{b}) = 0$.

Notice that we cannot distribute the dot product across the cross product like so: $\vec{a} \cdot (\vec{a} \times \vec{b}) = (\vec{a} \cdot \vec{b}) \times (\vec{a} \cdot \vec{b})$ since the dot product of two vectors is a scalar and the cross product is only defined for vectors.