Points P and Q have position vectors p = 2i − j - 3k and q = i + 4j - k. (a) Find the position vector of the midpoint M of [PQ]. (b) Point R lies on the line (PQ) such that QR = QM. Find the coordinates of R if R and M are distinct points.

Solve the equation graphically by finding x-intercepts. Confirm by using factoring to solve the equation.

$$x(3x-7)=6$$

Points A and B have position vectors

$$a = \left( \begin{array} { l } { 2 } \\ { 2 } \\ { 1 } \end{array} \right)$$

and

$$b = \left( \begin{array} { r } { 1 } \\ { - 1 } \\ { 3 } \end{array} \right)$$

Point C lies on [AB] so that AC : BC = 2 : 3. Find the position vector of C.

During a recent wet spell, the water level in Jasper’s Pond rose

$$2 \frac { 3 } { 4 }$$

in. The depth of the pond was then 10 ft 3 in. What was the depth of the water in the pond before the wet spell?

Compute

$$(T_0oT_2oT_1)(x, y).$$

$$T_1(x, y) = (x + y, y, -x), T_2(x, y, z) = (0, x + y + z, 3y), T_3(x, y, z) = (3x + 2y, 4z - x - 3y)$$

The size P of a certain insect population at time t (in days) obeys the model $P(t)=500 e^{0.02 t}$. Determine the number of insects at t = 0 days.

Find the possible values of the constant c such that the vector

$$\left( \begin{array} { c } { 2 c } \\ { c } \\ { - c } \end{array} \right)$$

has magnitude 12.

Define a relation R on Z as follows: For all integers m and n,

$$m R n \Leftrightarrow 5$$

|

$$\left( m ^ { 2 } - n ^ { 2 } \right)$$

. a. Is 1 R (-9)?, b. Is 2 R 13? , c. Is 2 R (-8)?, d. Is (-8) R 2?

Explain why it is impossible for a vector to have the given direction angles.

$$\alpha = 20 ^ { \circ } , \quad \beta = 45 ^ { \circ }$$

A shirt regularly prices at $40 is on sale for 25% off. Mentally calculate 25% of$40. Explain the method you used to arrive at your answer.

You pick a marble from a bag containing 1 green marble, 4 red marbles, 2 yellow marbles, and 3 black marbles. You replace the first marble and then select a second one. Find each probability. P(red, then yellow)

Evaluate each expression for s = 4 and t = 8.

$$( s t ) ^ { 2 }$$

Evaluate each expression for x=3, y=-1, and z=2.

$$x ^ { - y }$$

In "The Tragedy of Julius Caesar", how does the tyrant seize total power?