8. Vectors

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||u||.

Use the figure to find each vector: u - v. Use vector notation as in Example 4.

Vector v has the given direction angle and magnitude. Find the horizontal and vertical components.

θ = 27° 30' |v| = 15.4

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = 3i - 2j

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.

3v - 4w

Use the figure to find each vector: u - v. Use vector notation as in Example 4.

A ship leaves port on a bearing of 34.0° and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?

Given two pairs of vectors, $a$ and $b$, and $e$ and $f$, if $a\cdot b$ = $0$ and $e\cdot f$ = $0$, what can be concluded about the relationship between each pair of vectors?

Given vectors u and v, find: 2u.

u = 2i, v = i + j

Solve each problem. See Examples 5 and 6.

Bearing and Ground Speed of a Plane An airline route from San Francisco to Honolulu is on a bearing of 233.0°. A jet flying at 450 mph on that bearing encounters a wind blowing at 39.0 mph from a direction of 114.0°. Find the resulting bearing and ground speed of the plane.

Given vectors u and v, find: 2u + 3v.

u = 2i, v = i + j

Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.

forces of 250 and 450 newtons, forming an angle of 85°

Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.

-5v