1.5 The Cross Product (continued)

Object Types and Vector Notation

For each of the expressions below, state whether the result is a scalar, a vector, or undefined. You should write a sentence or two about each to explain your reasoning. (Assume that vectors are nonzero and not orthogonal.) -$\frac{\vv}{\vw}$-$\frac{\vv}{\vw\cdot\vv}$-$(\vu\times\vw)+\vv$-$k(\vu\cdot\vw)+c\vv$-$k(\vu\cdot\vv)\cdot\vw$-$\frac{\vv}{\vecmag{\vw}}$-$(k+\vu)\times \vv$-$k+(\vu\times \vv)$-$\vv+(\vu\cdot \vw)$-$k+\vecmag{\vu\times \vw}$

Use the operations of dot product and vector subtraction to write an expression involving $\vu$, $\vv$, and $\vw$ that evaluates to a scalar. You can use other operations if you want.

Use the operations of cross product, scalar multiplication, and vector addition to write an expression involving $\vu$, $\vv$, and $\vw$ that evaluates to a vector. You can use other operations if you want.

Use the operations of dot product and vector addition to write an expression involving $\vu$, $\vv$, and $\vw$ that is undefined. You can use other operations if you want.

In , we stated “Because the cross product works well with linear combinations, $\vu\times\vv$ is perpendicular to any linear combination of $\vu$ and $\vv$.” We want to verify this algebraically here. Specifically, show that $\vu\times\vv$ will be orthogonal to any vector of the form $a\vu+b\vv$.