91Ó°ÊÓ

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $$\mathbf{w}=\mathbf{i}-2 \mathbf{j}$$

Use vectors to find the interior angles of the triangle with the given vertices. $$(-3,5),(-1,9),(7,9)$$

A flagpole at a right angle to the horizontal is located on a slope that makes an angle of \(12^{\circ}\) with the horizontal. The flagpole's shadow is 16 meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is \(20^{\circ} .\) (a) Draw a triangle to represent the situation. Show the known quantities on the triangle and use a variable to indicate the height of the flagpole. (b) Write an equation that can be used to find the height of the flagpole. (c) Find the height of the flagpole.

Two ships leave a port at 9 A.M. One travels at a bearing of \(\mathrm{N} 53^{\circ} \mathrm{W}\) at 12 miles per hour, and the other travels at a bearing of \(\mathrm{S} 67^{\circ} \mathrm{W}\) at 16 miles per hour. Approximate how far apart they are at noon that day.

The initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j} .\) $$\begin{array}{cc}\text{Initial Point} && \text{Terminal Point} \\ (-2,1) && (3,-2) \end{array}$$

A triangular parcel of land has 115 meters of frontage, and the other boundaries have lengths of 76 meters and 92 meters. What angles does the frontage make with the two other boundaries?

The initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j} .\) $$\begin{array}{cc}\text{Initial Point} && \text{Terminal Point} \\ (-1,-5) && (2,3) \end{array}$$

Find the magnitude and direction angle of the vector v. $$\mathbf{v}=8\left(\cos 135^{\circ} \mathbf{i}+\sin 135^{\circ} \mathbf{j}\right)$$

Find the component form of \(v\) given its magnitude and the angle it makes with the positive \(x\) -axis. Sketch v. $$\begin{array}{cc}\text{Magnitude} && \text{Angle} \\ \|\mathbf{v}\|=3 && \theta=0^{\circ} \end{array}$$

A truck with a gross weight of 30,000 pounds is parked on a slope of \(d^{\circ}\) (see figure). Assume that the only force to overcome is the force of gravity. (a) Find the force required to keep the truck from rolling down the hill in terms of the slope \(d .\) (b) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|}\hline d & 0^{\circ} & 1^{\circ} & 2^{\circ} & 3^{\circ} & 4^{\circ} & 5^{\circ} \\\\\hline \text { Force } & & & & & & \\\\\hline\end{array}$$ $$\begin{array}{|l|l|l|l|l|l|}\hline d & 6^{\circ} & 7^{\circ} & 8^{\circ} & 9^{\circ} & 10^{\circ} \\\\\hline \text { Force } & & & & & \\\\\hline\end{array}$$ (c) Find the force perpendicular to the hill when \(d=5^{\circ}.\)