4 Linear Algebra I

1. Show the vectors
   
   $$\vec F= \sqrt{2} \hat\imath + \hat\jmath - 6 \hat k \qquad \vec G= 8 \hat\imath + 2 \hat k$$
   
   
   as neatly and accurately as possible in a 3D left-handed coordinate system with the $x$-axis to the right and the $z$ axis upwards in the plane of the paper. Now, using a different color, construct graphically, as accurately as possible, $2\vec F$, $3\vec G$, $\vec F+\vec G$, $\vec F-\vec G$ and also demonstrate them graphically in a neat plot, as accurately as possible. Compute these quantities also algebraically. Does that look about right in the graph? Compute $\vert\vert\vec F\vert\vert$ and $\vert\vert\vec G\vert\vert$. After evaluating the results in your calculator, do they agree with what you measure in the graph (ignoring that $\vec F$ is sticking a bit out of the paper)? Evaluate the angle between vectors $\vec F$ and $\vec G$ using a dot product. Does that angle seem about right? How would the above work out for the vectors
   
   $$\vec v= (\sqrt{2},1,-6) \qquad \vec w= (8,0,2)$$
   
   
   Double-check you are using the correct vectors while doing all the above. Make sure you have answered everything. As always, give exact, cleaned up, answers. As always, explain all reasoning! [2, chapter 6].

2. A flat mirror passes through the points
   
   $$\vec r_A=(-2,1,6) \quad \vec r_B=(2,1,-7) \quad \vec r_C= (4,2,1)$$
   
   
   Find a vector $\vec N$, with integer components, that is normal to the plane of the mirror using the appropriate vector combinations and product. Find a simple scalar equation for the plane of the mirror. Check that A, B, and C satisfy it. Now assume a laser beam moves in the positive $\vec{e}=\hat\imath$ direction before it is reflected by the mirror. Find the unit vector $\vec{e} {}'$ in the direction of the reflected beam. (Note: Snell's law of reflection may be formulated as follows: reflection leaves the component of $\vec{e}$ in the direction parallel to the mirror the same. However, it inverts the component normal to the mirror. So to get $\vec{e} {}'$, first consider the expression for the component of $\vec{e}$ in the direction normal to the mirror. You know that if $\vec n$ is a unit vector normal to the mirror, then this component is $\vec{e}\cdot\vec n$. The vector component is the scalar one multiplied by the unit vector $\vec n$. To invert this vector compont subtract it twice from $\vec{e}$. Once to zero the component and a second time to add the negative component. Therefore you see that:
   
   $$\vec e {}' = \vec e - 2 (\vec e \cdot \vec n) \vec n$$
   
   
   Do not compute $\vec n$ explicitly, as this would bring in a nasty square root. Just substitute $\vec n=\vec N/\vert\vec N\vert$ in the above expression.) You must use the procedures that require the least amount of algebra to answer all questions. Double-check you are using the correct vectors while doing all the above. Make sure you have answered everything. [2, chapter 6].

3. Given
   
   $$\vec r_A=(-2,1,6) \quad \vec r_B=(2,1,-7) \quad \vec r_C= (4,2,1) \quad \vec r_D=(2,2,2)$$
   
   
   Using appropriate vector products, find the area of the parallelogram with sides AB and AC, and the volume of the parallelepiped with sides AB, AC, and AD. Also find the angle between AD and the parallelogram. You must use the procedures that require the least amount of algebra to answer all questions. (To find the angle between a line and a plane, first find the angle between the line and a line normal to the plane, in the range from 0 to $\pi/2$ and then take the complement of that.) Double-check you are using the correct vectors while doing all the above. Make sure you have answered everything. [2, chapter 6].

4. Are the following sets of vector linearly independent, and why? Give the simplest rigorous reason.
   • $3\hat\imath +2\hat\jmath$ and $\hat\imath -\hat\jmath$.
   • $2\hat\imath$, $3\hat\jmath$, $5\hat\imath -12\hat k$, $\hat\imath +\hat\jmath +\hat k$.
   • $(1,0,0,0), (0,1,1,0), (-4,6,6,0)$
   [2, chapter 6].

5. Is the set S of vectors of the form $(x,y,2x,3y)$ in $R^4$ a vector space? Why? If so, give the dimension of the vector space and a basis. Also answer the same questions for the vector set $(x,x^2)$ and the vector set $(x,y,2+x,3+y)$. [2, chapter 6].

   Note: If for a vector in $R^n$ you define multiplication by a scalar $\alpha$ as
   

   $$\alpha \vec v \equiv \alpha (v_1, v_2, \ldots, v_n) \equiv (\alpha v_1, \alpha v_2, \ldots, \alpha v_n)$$
   
   
   and addition as
   
   $$\vec v + \vec w = (v_1, v_2, \ldots, v_n) + (w_1, w_2, \ldots, w_n) \equiv (v_1 + w_1, v_2 + w_2, \ldots, v_n+w_n)$$
   
   
   then the needed linear space properties, such as
   

   $$\alpha (\beta \vec v) = (\alpha \beta) \vec v$$ (1)

   $$\alpha (\vec v + \vec w) = \alpha \vec v + \alpha \vec w$$ (2)

   $$\vec v + \vec w = \vec w + \vec v$$ (3)

   $$\vec u + (\vec v + \vec w) = (\vec u + \vec v) + \vec w$$ (4)

   
   
   will be certainly satisfied. However, S is a subset of $R^4$ in which the vectors can all be written as $(x,y,2x,3y)$ for some values of $x$ and $y$. It is not obvious whether such a subset is complete under multiplication be a scalar or vector addition. In particular, if you take a scalar multiple of a vector $(x,y,2x,3y)$, is the result still of the form $(x',y',2x',3y')$ (with in general $x'\ne x$ and $y'\ne y$)? Or can the result no longer necessarily be written in a form $(x',y',2x',3y')$? In that case the result is no longer in S, so S is not complete under multiplication by a scalar. At least some multiples of vectors in S are outside S. The same for adding two vectors in S: is the result necessarily still in S?