The dot product

Consider the vectors

1. Find the lengths of the vectors, $\len{\vvec}$ and $\len{\wvec}$.

2. Find the dot product $\vvec\cdot\wvec$ and use it to find the angle $\theta$ between $\vvec$ and $\wvec$.

Consider the three vectors

1. Find the dot products $\uvec\cdot\uvec$, $\uvec\cdot\vvec$, and $\uvec\cdot\wvec$.

2. Use the dot products you just found to evaluate:

   1. $\len{\uvec}$.

   2. $(-5\uvec)\cdot\vvec$.

   3. $\uvec\cdot(-3\vvec+2\wvec)$.

   4. $\len{\frac1{\len{\uvec}} \uvec}$.

3. For what value of $k$ is $\uvec$ orthogonal to $k\vvec+5\wvec$?

Suppose that $\vvec$ and $\wvec$ are vectors where

1. What is $\len{\vvec}$?

2. What is the angle between $\vvec$ and $\wvec$?

3. Suppose that $t$ is a scalar. Find the value of $t$ for which $\vvec$ is orthogonal to $\wvec+t\vvec$?

Suppose that $\vvec=3\wvec$.

1. What is the relationship between $\vvec\cdot\vvec$ and $\wvec\cdot\wvec$?

2. What is the relationship between $\len{\vvec}$ and $\len{\wvec}$?

3. If $\vvec=s\wvec$ for some scalar $s$, what is the relationship between $\vvec\cdot\vvec$ and $\wvec\cdot\wvec$? What is the relationship between $\len{\vvec}$ and $\len{\wvec}$?

4. Suppose that $\vvec=\threevec{3}{-2}2$. Find a scalar $s$ so that $s\vvec$ has length 1.

Consider the vectors and a general vector $\xvec=\threevec xyz$.

1. Write an equation in terms of $x$, $y$, and $z$ that describes all the vectors $\xvec$ orthogonal to $\vvec_1$.

2. Write a linear system that describes all the vectors $\xvec$ orthogonal to both $\vvec_1$ and $\vvec_2$.

3. Write the solution set to this linear system in parametric form. What type of geometric object does this solution set represent? Indicate with a rough sketch why this makes sense.

4. Giv…

Explain your responses to these questions.

1. Suppose that $\vvec$ is orthogonal to both $\wvec_1$ and $\wvec_2$. Can you guarantee that $\vvec$ is also orthogonal to any linear combination $c_1\wvec_1+c_2\wvec_2$?

2. Suppose that $\vvec$ is orthogonal to itself. What can you say about $\vvec$?

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 apply the distributive property of dot products to explain why Find similar expressions for $c_2$ and $c_3$.

3. Verify that form a basis for $\real^3$ and that each vector is orthogonal to the other two.…

Suppose that $\vvec_1$, $\vvec_2$, and $\vvec_3$ are three nonzero vectors that are pairwise orthogonal; that is, each vector is orthogonal to the other two.

1. Explain why $\vvec_3$ cannot be a linear combination of $\vvec_1$ and $\vvec_2$.

2. Explain why this set of three vectors is linearly independent.

In the next chapter, we will consider certain $n\times n$ matrices $A$ and define a function where $\xvec$ is a vector in $\real^n$.

1. Suppose that $A=\begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix}$ and $\xvec=\twovec21$. Evaluate $q(\xvec) = \xvec\cdot(A\xvec)$.

2. For a general vector $\xvec=\twovec xy$, evaluate $q(\xvec) = \xvec\cdot(A\xvec)$ as an expression involving $x$ and $y$.

3. Suppose that $\vvec$ is an eigenvector of a matrix $A$ with associated eigenvalue $\lambda$ and that $\vvec$ has length 1. What is the value of the function $q(\vvec)$?