# Recent questions in Vectors and spaces

Vectors and spaces

### An observational study is retrospective if it considers only existing data. It is prospective if the study design calls for data to be collected as time goes on. Tell which of the following observational studies are retrospective and which are prospective. In a study of post-traumatic stress disorder, soldiers who have been in combat are given biannual physical and psychological tests for five years after they return from active duty.

Vectors and spaces

### Is there a relationship between Facebook use and age among college students? The following two-way table displays data for the 219 students who responded to the survey. \begin{array}{ccc} & \text{Age} \ \text{Facebook user?} & {\begin{array}{l|c|c|c} & \begin{array}{c} \text { Younger } \ (18-22) \end{array} & \begin{array}{c} \text { Middle } \ (23-27) \end{array} & \begin{array}{c} \text { Older } \ (28 \text { and up)} \end{array} \ \hline \text { Yes } & 78 & 49 & 21 \ \hline \text { No } & 4 & 21 & 46 \end{array}} \ \end{array} What percent of the students who responded were older Facebook users?

Vectors and spaces

### The reduced row echelon form of the augmented matrix of a system of linear equations is given. Tell whether the system has one solution, no solution, or infinitely many solutions. Write the solutions or, if there is no solution, say the system is inconsistent.

Vectors and spaces

### if x, y belong to $$R^p$$, than is it true that the relation norm $$(x+y) = norm (x) + norm (y)$$ holds if and only if $$x = cy\ or\ y = cx\ with\ c>0$$

Vectors and spaces

### Find the principal unit normal vector to the curve at the specified value of the parameter $$r(t)=ti+\frac{6}{t}j , t=3$$

Vectors and spaces

### Find $$|u \cdot v|$$ and determine whether $$u \cdot v$$ is directed into the screen or out of the screen. $$|u \cdot v| = ??$$

Vectors and spaces

### Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance $$\displaystyle\sigma^{{{2}}}$$ and (b) the population standard deviation $$\displaystyle\sigma$$. Interpret the results. The final exam scores of 24 randomly selected students in a statistics class are shown in the table at the left. Use a 95% level of confidence.

Vectors and spaces

### Find the vectors T, N, and B at the given point. $$r(t) =$$ and point $$<4,-\frac{16}{3},-2>$$

Vectors and spaces

### A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large. Questionnaires were sent to 200 randomly selected businesses of each size. Because not all questionnaires were returned, researchers decided to investigate the relationship between the response rate and the size of the business. The data are given in the two-way table. $$\begin{array} {lc} & \text{Business size} \ \text {Response?} &\begin{array}{l|c|c|c|c} & \text { Small } & \text { Medium } & \text { Large } & \text { Total } \ \text { Yes } & 125 & 81 & 40 & 246 \\ \hline \text { No } & 75 & 119 & 160 & 354 \\ \hline \text { Total } & 200 & 200 & 200 & 600 \\ \hline \end{array} \end{array}$$  Which variable should be used as the explanatory variable? Explain.

Vectors and spaces

### Lee buys a parcel of land in the shape of a rectangle. It has an area of 1,125 square feet and a width of 25 feet. What is the length of the parcel of land?

Vectors and spaces

### Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) $$x+2y-z=2,$$ $$2x-2y+z=1$$

Vectors and spaces

### Determine whether the following statement is true or false. A system of linear equations Ax=b has the same solutions as the system of linear equations Rx=c, where [R c] is the reduced row echelon form of [A b].

Vectors and spaces

### Vector Cross Product Let vectors $$A=(1,0,-3), B =(-2,5,1),\ and\ C =(3,1,1)$$. Calculate the following, expressing your answers as ordered triples (three comma-separated numbers). (C) $$(2\bar B)(3\bar C)$$ (D) $$(\bar B)(\bar C)$$ (E) $$\overrightarrow A(\overrightarrow B \times \overrightarrow C)$$ (F)If $$\bar v_1 \text{ and } \bar v_2$$ are perpendicular, $$|\bar v_1 \times \bar v_2|$$ (G) If $$\bar v_1 \text{ and } \bar v_2$$ are parallel, $$|\bar v_1 \times \bar v_2|$$

Vectors and spaces

### The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. [1−26\000]

Vectors and spaces

### The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. $$\begin{bmatrix}1 & -4&5 \\0 & 0&0 \\0 & 0&0 \\ \end{bmatrix}$$

Vectors and spaces

### A college student earned $8900 during summer vacation working as a waiter in a popular restaurant. Part was invested at 10% simple interest and the remainder at 6% simple interest. At the end of one year, the student had earned$674 interest. How much was invested at 10%?

Vectors and spaces

### Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR Consider the points below P(1,0,1) , Q(-2,1,4) , R(7,2,7) a) Find a nonzero vector orthogonal to the plane through the points P,Q and R b) Find the area of the triangle PQR

Vectors and spaces

### Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations. $$\begin{cases} 7 x-5 y=11\\ x-y=-5 \end{cases}$$

Vectors and spaces