A scalar is only a magnitude with appropriate units.

A vector has both direction and magnitude.

Two vectors that are parallel may be added or subtracted to find the resultant.

We describe vectors using the x,y-coordinate system.

If vectors are at right angles to each other, the magnitude of the resultant may be found using the Pythagorean theorem.

Trigonometry allows the proportions of similar right triangles to be compared.

The ratio of opposite to hypotenuse is called sine. The ratio of adjacent to hypotenuse is called cosine. The ratio of opposite to adjacent is called tangent.

Chief SOHCAHTOA

Example 1: If a cow in a pasture walks 120 meters at an angle 20° north of east, (A) how far has the cow moved to the north? (B) How far has the cow moved to the east?

Example 2: A man climbs a stairway and travels 4 meters horizontally and 3 meters vertically. What is the magnitude and direction of his displacement?

Example 3: A hiker climbs a hill and travels 55 meters horizontally and 65 meters vertically. What is the magnitude and direction of his displacement?

A single vector can be resolved (separated) into its components. This is done using the sine and cosine functions.

Example 4: Find the component velocities of a rocket traveling 110 m/s at an angle of 65° to the ground.

If vectors are not perpendicular, they may be added if each vector is resolved into components.

Example 5: Taylor runs 10 km in a direction 30° south of east. He then runs 15 km due north. Find the magnitude and direction of his resultant displacement.

Objects thrown or launched and subject to gravity are called projectiles. This type of motion is called projectile motion.

Projectiles follow parabolic trajectories.

Projectile motion is free fall with an initial horizontal velocity.

Example 6: A bridge is 230 m above the river. A baseball is thrown horizontally off the bridge at a speed of 60 m/s. What horizontal distance does it travel before it hits the water?

Example 7: A baseball thrown horizontally off the same bridge (230 m high) travels 110 meters in a horizontal direction before hitting the water. What was the velocity of the throw?

Example 8: The Royal Gorge Bridge is 321 m above the Arkansas River. A rock kicked horizontally off the bridge hits the water such that the horizontal displacement is 45.0 m. At what speed was the rock kicked?

Example 9 : Adam kicks the football at an angle of 40° to horizontal. If the initial velocity is 30 m/s, how far does the ball travel horizontally before it hits the ground?

Example 10: A hunter aims at a monkey hanging from a tree. The hunter is 30 m from a point on the ground directly below the monkey. The tip of the gun is a vertical distance of 40 m below the monkey. If the monkey lets go and falls as the gun is fired and the bullet travels 200 m/s, how much must the hunter lead the monkey to hit him?

All motion is relative to an arbitrarily selected frame of reference. We can express motion of an object relative to any frame of reference.

Example 11: A swimmer swims across a wide river with a velocity of 5.00 m/s south relative to the water. The water flows 7.00 m/s east. What is the swimmer’s velocity with respect to the shore?