11.2 Vectors in the Plane

Two Dimensional Vector Two dimensional vector v: 𝑎, 𝑏 where a and b are the components of the vector v. Standard representation: arrow from origin to 𝑎, 𝑏 Magnitude of v: length of arrow, 𝐯 = 𝑎 2 + 𝑏 2 Zero vector: 𝟎= 0,0 , has zero length and no direction Direction angle: smallest nonnegative angle θ formed with the positive x-axis and arrow

Head Minus Tail (HMT) Rule If an arrow has an initial point 𝑥 1 , 𝑦 1 and terminal point 𝑥 2 , 𝑦 2 , it represents the vector 𝑥 2 − 𝑥 1 , 𝑦 2 − 𝑦 1 .

Example 1 Find the component form of, magnitude of, and direction angle of the vector whose initial point is (1, 3) and terminal point is (2, -1).

Example 2 Find the component form of a vector with magnitude 3 and direction angle 40°.

Vector Addition and Scalar Multiplication

Example 3 Let u = −1, 3 and v = 4, 7 . Find the following: 2u + 3v 1 2 𝐮

Example 4 An airplane flying due east at 500 mph in still air encounters a 70 mph tail wind acting in the direction 60° north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

Example 5 A particle moves in the plane so that its position at any time 𝑡≥0 is given by sin 𝑡 , 𝑡 2 2 . Find the position vector of the particle at time t. Find the velocity vector of the particle at time t. Find the acceleration of the particle at time t. Describe the position and motion of the of the particle at time t = 6.

Velocity, Speed, Acceleration, and Direction of Motion Suppose a particle moves along a smooth curve in the plane so that its position at any time t is 𝑥 𝑡 , 𝑦(𝑡) , where x and y are differentiable functions. Position vector: 𝐫 𝑡 = 𝑥 𝑡 , 𝑦(𝑡) Velocity vector: 𝐯 𝑡 = 𝑑𝑥 𝑑𝑡 , 𝑑𝑦 𝑑𝑡 Speed: magnitude of velocity vector, 𝐯 Acceleration vector: 𝐚 𝑡 = 𝑑 2 𝑥 𝑑 𝑡 2 , 𝑑 2 𝑦 𝑑 𝑡 2 Direction of motion is the direction vector: 𝐯 𝐯 (also a unit vector)

Position Function of a Projectile Neglecting air resistance, the path of a projectile launched from an initial height h with initial speed v0 and angle of elevation θ is described by the vector function Where g is the gravitational constant (32 ft/sec2 or 9.81 m/sec2)

Example 6 A baseball leaves a batters bat at an angle of 60° from the horizontal, traveling at 110 feet per second. A fielder, who is standing in the direct line of the hit with is glove at a vertical reach of 8 feet, catches the ball for the out. His feet do not leave the ground. Find the position vector of the baseball at time t (in seconds). Find the velocity vector of the baseball at time t (in seconds). At what time t does the fielder catch the ball, and how far is he from home plate? How fast is the ball traveling when he catches the ball?

Example 7 A particle moves in the plane with position vector 𝐫 𝑡 = sin 3𝑡 , cos⁡(5𝑡) . Find the velocity and acceleration vectors and determine the path of the particle.

Example 8 A particle moves in an elliptical path so that its position at any time 𝑡≥0 is given by 4 sin 𝑡,2 cos 𝑡 . Find the velocity and acceleration vectors. Find the velocity, acceleration, speed, and direction of motion at 𝑡= 𝜋 4 Sketch the path of the particle and show the velocity vector at the point (4, 0). Does the particle travel clockwise or counterclockwise around the origin?

Displacement and Distance Traveled Suppose a particle moves along a path in the plane so that its velocity at any time t is 𝐯 𝑡 = 𝑣 1 𝑡 , 𝑣 2 (𝑡) where 𝑣 1 and 𝑣 2 are integrable functions of t. The displacement from t = a and t = b is given by the vector 𝑎 𝑏 𝑣 1 𝑡 𝑑𝑡, 𝑎 𝑏 𝑣 2 𝑡 𝑑𝑡 The preceding vector is added to the position at time t = a to get the position at t = b. The distance traveled from t = a to t = b is 𝑎 𝑏 𝐯(𝑡) 𝑑𝑡= 𝑎 𝑏 𝑣 1 (𝑡) 2 + ( 𝑣 2 𝑡 ) 2 𝑑𝑡

Example 9 A particle moves in the plane with velocity vector 𝑣 𝑡 = 𝑡−3𝜋 cos 𝜋𝑡, 2𝑡−𝜋 sin 𝜋𝑡 . At t = 0, the particle is at the point (1, 5). Find the position of the particle at t = 4. What is the total distance traveled by the particle from t = 0 to t = 4?