vectors Precalculus

Vectors A vector is an object that has a magnitude and a direction. Given two points P: & Q: on the plane, a vector v that connects the points from P to Q is (start with last point) Notations v = i + j. Or The magnitude of v is |v| = The direction is the slope Vectors are equal if they have the same magnitude and the same direction

example Find the component vector for r= (3,8) s= (-4,5) Find the magnitute of Find the direction of

Vectors are equal if they have the same magnitude and the same direction Find the component vector for r= (3,8) s= (-4,5) Find the magnitute of The direction of

Unit Vectors Unit vectors are vectors of length 1. i is the unit vector in the x direction. j is the unit vector in the y direction. A unit vector in the direction of v is v/||v|| A vector v can be represented in component form by v = v x i + v y j. The magnitude of v is ||v|| = The unit vector is

Vector Operations Scalar multiplication: A vector can be multiplied by any scalar (or number). Example: Let v = 5i + 4j, k = 7. Then kv = 7(5i + 4j) = 35i + 28j. Addition/subtraction of vectors: Add/subtract same components. Example Let v = 5i + 4j, w = –2i + 3j. v + w = (5i + 4j) + (–2i + 3j) = (5 – 2)i + (4 + 3)j = 3i + 7j.

Example: 3v – 2w = ||3v – 2w|| =

Example: 3v – 2w = ||3v – 2w|| =

Direction Angles Given the direction angle of a vector, find the component form of the vector in the same direction. Use the formula: Use the formula above for number 29 on pg. 511

Direction Angles Find the direction angle of the vectors Start by plotting the vector. What trig function will help find the angle?

Direction Angles Find the direction angle of the vectors But w is in quadrant II, so =123.7

Direction Angles Find the direction angle of a vector, then find the component form of the vector with magnitude of 6 in the same direction. Ex:

Direction Angles Given the direction angle of a vector, find the component form of the vector of magnitude 6 in the same direction. Use the formula: Ex:

Dot Product Dot Product: Multiplication of two vectors. Let v = v x i + v y j, w = w x i + w y j. v · w = v x w x + v y w y Example: Let v = 5i + 4j, w = –2i + 3j. v · w = (5)(–2) + (4)(3) = – = 2** **vectors v and w are orthogonal (perpendicular) iff v · w = 0.

Orthogonal, parallel vectors Do now: find cos 90 degrees If the dot product = 0 the vectors are orthogonal If the direction (slope) of the vectors is the same, the vectors are parallel examples: Are the following vectors pairs orthogonal or parallel or neither?

Orthogonal, parallel vectors If the dot product = 0 the vectors are orthogonal If the direction (slope) of the vectors is the same, the vectors are parallel Examples: Are the following vectors pairs orthogonal or parallel or neither? U and v are orthogonal W and z are not orthogonal so check direction: Direction of w = -5/3 and direction of z = -5/3 so they are parallel!

Alternate Dot Product Alternate Dot Product formula: v · w = ||v||||w||cos(θ). The angle θ is the angle between the two vectors. θ V W

Angles between 2 vectors Using the alternate formula, we solve for θ: v · w = ||v||||w||cos(θ). The angle is between 0 and 180 degrees

Example v = 5i + 4j, w = –2i + 3j. The angle is between 0 and 180 degrees

Example v = 5i + 4j, w = –2i + 3j. They are neither orthogonal nor parallel so we find the angle…

Example v = 5i + 4j, w = –2i + 3j. The angle is between 0 and 180 degrees

Student will be able to solve problems involving velocity and other quantities that can be represented by vectors Relation to real life: The Malaysian plane that is lost( in the recent news) could be tracked with calculations of velocity. This is where they started when looking for the wreckage. Vocabulary: Bearing with respect to Navigation-coming out of the north, measured clockwise Magnitude: Length Velocity: (has magnitude and direction) Speed: the magnitude of velocity is speed.

Real life problem Components of a vector (gives east and north speeds) Problem: An airplane is flying on a bearing of 170 o at 460mph. Find the component form of the velocity of the airplane.

Bearing

Wind vectors Next problem: An airplane is flying on a bearing of 340 o at 325 mph. A wind is blowing with the bearing of 320 o at 40 mph. Find the component form of the velocity of the plane and the wind. Then find the actual speed and direction.

Calculating two vectors-1 st : velocity vector: Now Calculate the wind vector

Wind vector:

Wind vector added for the wind vector, 320 degrees corresponds to 130 degrees: so the actual velocity =