Activity 6.3.2

Consider the vectors $$\wvec_1 = \threevec1{-1}1,\hspace{24pt} \wvec_2 = \threevec1{1}0,\hspace{24pt} \wvec_3 = \threevec1{-1}{-2}.$$

1. Verify that this set forms an orthogonal set of $3$-dimensional vectors.

2. Explain why we know that this set of vectors forms a basis for $\real^3$.

3. Suppose that $\bvec=\threevec24{-4}$. Find the weights $c_1$, $c_2$, and $c_3$ that express $\bvec$ as a linear combination $\bvec=c_1\wvec_1 + c_2\wvec_2 + c_3\wvec_3$ using this proposition.

4. If we multiply a vector $\vvec$ by a positive scalar $s$, the length of $\vvec$ is also multiplied by $s$; that is, $\len{s\vvec} = s\len{\vvec}$.

   Using this observation, find a vector $\uvec_1$ that is parallel to $\wvec_1$ and has length 1. Such vectors are called unit vectors.

5. Similarly, find a unit vector $\uvec_2$ that is parallel to $\wvec_2$ and a unit vector $\uvec_3$ that is parallel to $\wvec_3$.

6. Construct the matrix $Q=\begin{bmatrix} \uvec_1 & \uvec_2 & \uvec_3 \end{bmatrix}$ and find the product $Q^TQ$. Use this proposition to explain your result.