Exam 12: Vectors and Matrices

Given $\mathbf{R}=\left(\begin{array}{cc}-4 & -4 \\ 1 & 5\end{array}\right), \mathbf{S}=\left(\begin{array}{cc}1 & -5 \\ -2 & 7\end{array}\right), \vec{u}=(1,1)$ , and $\vec{v}=(5,2)$ , does $(\vec{u} \cdot \vec{v})(\mathbf{R}+\mathbf{S})=\left(\begin{array}{cc}-3 & -9 \\ -5 & 24\end{array}\right) ?$

A country has two main political parties. The vector $\vec{P}=(f, s, n)$ gives the number of people who are members of the first party, the second party, and neither party, respectively. Suppose $\vec{P}_{\text {new }}=\mathbf{T} \vec{P}_{\text {old }}=\left(\begin{array}{lll}0.85 & 0.03 & 0.10 \\ 0.11 & 0.95 & 0.15 \\ 0.04 & 0.02 & 0.75\end{array}\right) \vec{P}_{\text {old }}$ . Let $\vec{P}_{2005}=(22,20,14)$ be the population vector (in tens of thousands) in the year 2005. What is $\vec{P}_{2007}$ ?

If $R=(2,3,4,5,6)$ and $S=(0,1,3,4,7)$ , then what is $\vec{p}=\frac{\vec{R}}{2}+\frac{4 \vec{S}}{4}$ ?

Given that $\mathbf{A}=\left(\begin{array}{ccc}1 & -2 & 2 \\ 4 & -1 & -3 \\ 4 & 0 & 2\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{ccc}0 & 1 & -3 \\ 2 & -4 & 0 \\ -1 & -2 & 3\end{array}\right)$ , let $2 \mathbf{A}-2 \mathbf{B}=\mathbf{C}$ , where $\mathbf{C}=\left(\begin{array}{lll}c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33}\end{array}\right)$ . What is $c_{11} ?$

Is the distance between a satellite and the earth a vector or a scalar?

A country has two main political parties. The vector $\vec{P}=(f, s, n)$ gives the number of people who are members of the first party, the second party, and neither party, respectively. Suppose $\vec{P}_{\text {new }}=\mathbf{T} \vec{P}_{\text {old }}=\left(\begin{array}{ccc}0.90 & 0.11 & 0.10 \\ 0.09 & 0.85 & 0.15 \\ 0.01 & 0.04 & 0.75\end{array}\right) \vec{P}_{\text {old }}$ . Which party's members are the most loyal?

The vector starting at the point $P=(2,4)$ and ending at the point $Q=(7,5)$ can be resolved into the components ----------------- $\vec{i}+$ -------------- $\vec{j}$

A retailer's total monthly sales of three different models of television is given by the vector $\vec{S}=(14,27,20)$ . If the sales for each model go up by 7 the next month, what is $\vec{Q}$ , the next month's total sales?

A horse runs at a constant speed of $18 \mathrm{~m} \mathrm{sec}$ . He starts at a fence and his path makes an angle of $13^{\circ}$ with the fence. After 9 seconds, how far is he from the fence? Round numbers to 3 decimal places if necessary.

A particle in equilibrium is acted upon by three forces, two of which have components $4 \vec{i}+6 \vec{j}-9 \vec{k}$ and $9 \vec{i}-4 \vec{j}+3 \vec{k}$ . The components of the third must be ------ $\underline{\vec{i}}+\ldots \vec{j}+\ldots \vec{k}$ .

Perform the computation $(5 \vec{i}-2 \vec{j})+(5 \vec{i}+4 \vec{j})$ .

In the figure below, the angle between the $x$ -axis and the vector $\overrightarrow{R S}$ is------------° Round to the nearest whole number.

Simplify the following: $2(6 \vec{v}-\vec{w}-\vec{p})+9(\vec{v}-4 \vec{w}+\vec{p})$

Let $\mathbf{S}=\left(\begin{array}{cccc}1 & 0 & -1 & -2 \\ 2 & 1 & 0 & 1 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & 3 & 2\end{array}\right), \vec{u}=(-3,1,-1,0)$ , and $\vec{v}=(1,2,0,-3)$ . What is $\mathbf{S} \vec{u} \cdot \mathbf{S} \vec{v}$ ?

A particle is acted on by two forces, one of them to the west and of magnitude 0.5 dynes (a dyne is a unit of force), and the other in the direction $60^{\circ}$ north of east and of magnitude 1 dyne. A third force acting upon the particle that would keep it at equilibrium has a magnitude of ------- dynes and points ------------(north \ south \ east \ west). Round the first answer to 2 decimal places.

Let $P=(-1,3)$ and $Q=(2,4)$ . Find a vector of length 9 pointing in the opposite direction of $\overrightarrow{P Q}$ .

A model pyramid is built using four equilateral triangles connected to a square base. If the length of one side of the base is 11 inches, how many inches high is the pyramid? Round to 2 decimal places.

Let $\vec{v}$ be a vector of length 3 pointing $30^{\circ}$ north of east. Find the length and direction of $23 \vec{v}$ .

An airplane is flying at an airspeed of 600 km/hr in a crosswind blowing from the southeast at a speed of 50 km/hr. To end up going due west, the plane should head ------------° south of west and will have a speed of ----------- km/hr relative to the ground. Round each answer to 2 decimal places.

Simplify $2(4 \vec{v}+6 \vec{w})+\vec{v}$