4 Linear Algebra I

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

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, 7.1-2].

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. Hints: $\vec r_B-\vec r_A$ and $\vec r_C-\vec r_a$ are two vectors in the plane, and you need a vector normal to the plane. How would you get it? Double-check you are using the correct vectors while doing all the above. Make sure you have answered everything. [2, 7.1-2,5 ex. 9].

3. Continuing the previous question, assume that 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. Make a picture of the reflection. Hints: Snell's law of reflection may be formulated as follows: reflection leaves the component, [2, 7.3, ex. 6-7], 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 $\vec{e}\cdot\vec n$, [2, 7.3, ex. 6]. The vector component is the scalar one multiplied by the unit vector $\vec n$, [2, 7.3, ex. 7]. To invert this vector component 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.

4. 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 triangle with sides AB and AC, 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 vector 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, 7.4].

5. 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, 7.6].

6. 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 $(x,y,2+x,3+y)$.

   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.

   The problem is to show that the vector sets S are complete. For a vector space, a multiple of a vector must still be in the same space. So must the sum of any two vectors still be in the space. Otherwise the space is not complete. In the first potential vector space S above, if you multiply a vector of the form $(x,y,2x,3y)$ by some constant, you get another vector. That other vector is only part of S if it can be written as $(x',y',2x',3y')$ for some values $x'$ and $y'$. If it cannot be written in this form, the vector is not part of S, and so S is not a vector space. The same for adding two vectors in S. If you add two vectors $(x_1,y_1,2x_1,3y_1)$ and $(x_2,y_2,2x_2,3y_2)$, is the result of the form $(x_3,y_3,2x_3,3y_3)$ for some $x_3$ and $y_3$? Otherwise the sum of two vector in S is not in S and S is not a vector space. [2, 7.6].