Then/Now You used trigonometry to solve triangles. (Lesson 5-4) Represent and operate with vectors geometrically. Solve vector problems and resolve vectors into their rectangular components.

Vocabulary vector initial point terminal point standard position direction magnitude quadrant bearing true bearing parallel vectors equivalent vectors opposite vectors resultant triangle method parallelogram method zero vector components rectangular components

Example 1 Identify Vector Quantities A. State whether a hockey puck shot northwest at 60 miles per hour is a vector quantity or a scalar quantity. Vector – a quantity that has both magnitude (length) and direction. B. State whether a tennis ball served at 110 miles per hour is a vector quantity or a scalar quantity. C. State whether a sprinter running 100 meters north is a vector quantity or a scalar quantity.

Example 2 Represent a Vector Geometrically A. Draw an arrow diagram for v = 10 newtons of force at 30° to the horizontal. B. Draw an arrow diagram for z = 25 meters per second at a bearing of S70°W.

Example 2 Represent a Vector Geometrically D. Draw an arrow diagram for t = 10 miles per hour at a bearing of 025°. C. Draw an arrow diagram for z = 150 meters per second at a bearing of W45°N.

Key Concept 3

Example 4 Operations with Vectors Draw a vector diagram of a + b

Example 4 Operations with Vectors Draw a vector diagram of a – 3b.

Example 4 Draw a vector diagram of n + 2m. A. B. C. D.

Example 5 ROWING Jamie rows her boat due east at a speed of 20 feet per second across a river directly toward the opposite bank. At the same time, the current of the river is carrying her due south at a rate of 4 feet per second. Find Jamie’s speed and direction relative to the shore.

Example 5 Use Vectors to Solve Navigation Problems AVIATION An airplane is flying with an airspeed of 475 miles per hour on a heading of 070°. If an 80- mile-per-hour wind is blowing from a true heading of 120°, determine the velocity and direction of the plane relative to the ground.

Example 6 A. GARDENING While digging in his garden, Will pushes a shovel into the ground with a force of 630 newtons at an angle of 70° with the ground. Draw a diagram that shows the resolution of the force that Will exerts into its rectangular components. Resolve a Force into Rectangular Components

Example 6 B. GARDENING While digging in his garden, Will pushes a shovel into the ground with a force of 630 newtons at an angle of 70° with the ground. Find the magnitudes of the horizontal and vertical components of the force. The horizontal and vertical components of the force form a right triangle. Use the sine or cosine ratios to find the magnitude of each force. Resolve a Force into Rectangular Components

Example 6 SOCCER A player kicks a soccer ball so that it leaves the ground with a velocity of 39 feet per second at an angle of 37° with the ground. Find the magnitude of the horizontal and vertical components of the velocity. A.horizontal component ≈ ft/s; vertical component ≈ ft/s B.horizontal component ≈ 37 ft/s; vertical component ≈ 39 ft/s C.horizontal component ≈ ft/s; vertical component ≈ ft/s D.horizontal component ≈ ft/s; vertical component ≈ ft/s