Q. 18 - Calculus

Question

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

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

Step-by-Step Solution

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 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.