Exercise 8

Suppose that $\vvec_1$, $\vvec_2$, and $\vvec_3$ form a basis for $\real^3$ and that each vector is orthogonal to the other two. Suppose also that $\vvec$ is another vector in $\real^3$.

1. Explain why $\vvec=c_1\vvec_1+c_2\vvec_2+c_3\vvec_3$ for some scalars $c_1$, $c_2$, and $c_3$.

2. Beginning with the expression $$\vvec\cdot\vvec_1 = (c_1\vvec_1+c_2\vvec_2+c_3\vvec_3)\cdot\vvec_1,$$ apply the distributive property of dot products to explain why $$c_1=\frac{\vvec\cdot\vvec_1}{\vvec_1\cdot\vvec_1}.$$ Find similar expressions for $c_2$ and $c_3$.

3. Verify that $$\vvec_1=\threevec121,\hspace{24pt} \vvec_2=\threevec1{-1}1,\hspace{24pt} \vvec_3=\threevec10{-1}$$ form a basis for $\real^3$ and that each vector is orthogonal to the other two. Use what you've discovered in this problem to write the vector $\vvec=\threevec35{-1}$ as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$.