Practice Exam 1 for 18.02, Spring 1999

Problem 1 (25 points; 10, 10, 5) Let $\bf A = i+2j + 2k$, $\bf B=i+j+k$.

1.

Find a vector perpendicular to both $\bf A$ and $\bf B$.

2.

Find the equation of the plane passing through the point$(1,0,8)$ and parallel to both $\bf A$ and $\bf B$.

3.

Determine where the plane meets the $y$-axis.

Solution

1.

${\bf N}={\bf A}\times{\bf B}=j-k$ is perpendicular to both.

2.

The plane is ${\bf N}\cdot{\bf x}=d$ so $y-z=-8.$

3.

It meets the $y$-axis (which is $x=0,z=0)$ at the point $(0,-8,0).$

Problem 2 (15 points) Use vector methods to show that the line joining the mid-points of two sides of a triangle is parallel to the third side.

Solution If the vertices of the triangle are $P,$ $Q$ and $R$ then the sides are ${\bf PQ},$ ${\bf QR}$ and ${\bf RP}.$ The vector from the midpoint of PQ to the midpoint of $QR$ is $\frac12({\bf PQ}+{\bf QR}).$ Summing the three sides gives zero so ${\bf PQ}+{\bf QR}+{\bf RP}=0$ which implies that $\frac12({\bf PQ}+{\bf QR})=-\frac12{\bf RP},$ so the line between the midpoints of two sides is parallel to the base ${\bf RP}.$

Problem 3 (25 points; 20, 5) Consider the system of linear equations

$$2x_1 +3x_2 +cx_3 = y_1$$

$$-x_1 + x_3 = y_2$$

$$x_1 + x_2 + x_3 = y_3$$

1.

Take $c=3$. Write the system in matrix form $\bf Ax=y$. Calculate $\bf {A}^{-1}$ and use it to find equations expressing $x_1$, $x_2$, $x_3$ in terms of $y_1$, $y_2$ and $y_3$.

2.

For what value(s) of $c$ is it not possible to solve for the $x$'s in terms of the $y$'s?

Solution

1.

The matrix ${\bf A}= \left[\begin{matrix}2&3&c\\ -1&0&1\\ 1&1&1\end{matrix}\right].$ For $c=3$ the cofactor matrix of this is $\left[\begin{matrix}-1&2&-1\\ 0&-1&1\\ 3&-5&3\end{matrix}\right].$ The determinant (in general) is $3-c+3-2=4-c=1$ when $c=3.$ Thus the inverse matrix is $\left[\begin{matrix}-1&0&3\\ 2&-1&-5\\ -1&1&3\end{matrix}\right].$ It follows that ${\bf x}={\bf A}\cdot{\bf y}$ so

$$x_1=-y_1+3y_3,\ x_2=2y_1-y_2-5y_3,\ x_3=-y_1+y_2+3y_3.$$

2.

For $c=4$ since the determinant of ${\bf A}$ vanishes and there is then either no solution or an infinite number of solutions depending on ${\bf y}.$

Problem 4 (25 points; 10, 5, 5, 5) The motion of a point $P=(x,y,z)$ in space is described by the parametric equations

$$x=2+t^2 \, \quad y=t+1 \, \quad z=t^2+4t+1 \, .$$

1.

Does the curve meet the plane $x+z=0$?

2.

Where does the curve meet the plane $y=0$?

3.

Compute the velocity vector for the curve.

4.

Find the point at which the speed is smallest.

Solution

1.

On the curve $x+z=2t^2+4t+3=2(t+1)^2+1$ which never vanishes, so the curve does not meet the plane $x+z=0.$

2.

The velocity vector is $2t\hat i+\hat j+(2t+4)\hat k.$

3.

The square of the speed is $4t^2+1+4(t+2)^2=8(t+1)^2+9$ so the minimum occurs at $t=-1,$ which means at the point $(3,0,-2).$

Problem 5 (10 points) Consider the vectors

$$\bf A=i-j+k \, \quad B=i+j \, \hbox{ and } C=i-j-2k \, .$$

1.

Show that each is perpendicular to the other two.

2.

Find constants $c_1$, $c_2$, $c_3$ so that ${\bf i}=c_1{\bf A}+c_2{\bf B} + c_3{\bf C}$.

Solution

1.

${\bf A\cdot B}=1-1=0,$ ${\bf B\cdot C}=1-1=0,$ ${\bf A\cdot C}=1+1-2=0$ so each is perpendicular to the others.

2.

The dot products are ${\bf i\cdot A}=1,$ ${\bf i\cdot B}=1,$ ${\bf i\cdot C}=1.$ The squares of the lengths are $\vert{\bf A}\vert^2=3,$ $\vert{\bf B}\vert^2=2$ and $\vert{\bf C}\vert^2=6$ so

$${\bf i}=\frac13{\bf A}+\frac12{\bf B}+\frac16{\bf C},\ c_1=\frac13,\ c_2=\frac12,\ c_3=\frac16.$$