Question

If u, v and w are three vectors in ${\mathrm{ℝ}}^3$, which of the following products make sense and which do not?

localid="1649346164463" $$\begin{gathered} \left(a\right)u·(v·w) \\ \left(b\right)u·(v\times w) \\ \left(c\right)u\times (v·w) \\ \left(d\right)u\times (v\times w) \end{gathered}$$

Short answer

$$\begin{gathered} \left(a\right)u·(v·w)\mathrm{does}\mathrm{not}\mathrm{make}\mathrm{sense}. \\ \left(b\right)u·(v\times w)\mathrm{make}\mathrm{sense}. \\ \left(c\right)u\times (v·w)\mathrm{does}\mathrm{not}\mathrm{make}\mathrm{sense}. \\ \left(d\right)u\times (v\times w)\mathrm{make}\mathrm{sense}. \end{gathered}$$

Step-by-step solution

Step 1. Given Information 

If u, v and w are three vectors in ${\mathrm{ℝ}}^3$, which of the following products make sense and which do not?

$$\begin{gathered} \left(a\right)u·(v·w) \\ \left(b\right)u·(v\times w) \\ \left(c\right)u\times (v·w) \\ \left(d\right)u\times (v\times w) \end{gathered}$$

Part (a) Step 1. The first product is u·(v·w)

The combination$u·(v·w)$is not defined, since u·v is a scalar and we

need two vectors to form a dot product.

Part (b) Step 1. The first product is u·(v×w)

The product $u·(v\times w)$is defined and has an important geometrical interpretation. This is examples of triple scalar products.

Part (c) Step 1. The first product is u×(v·w)

The product $u\times (v·w)$does not make sense because $v·w$ is a scalar which cannot be crossed with a vector.

Part (d) Step 1. The first product is u×(v×w)

The product $u\times (v\times w)$make sense because $v\times w$is a vector which is crossed by a vector.