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VECTORS

This article explains the concept of vectors and scalars, their properties, and operations related to them. It covers topics such as magnitude, direction, collinearity, orthogonality, and manipulation of vectors. Helpful examples and exercises are provided.

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VECTORS

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  1. VECTORS Before you came along we had no direction!!

  2. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector. Blue and orange vectors have same magnitude but different direction. Blue and purple vectors have same magnitude and direction so they are equal. Blue and greenvectors have same direction but different magnitude. Two vectors are equal if they have the same direction and magnitude (length). Another word for length is NORM

  3. Vectors versus Scalar A vector is a quantity that has both magnitude and direction. A scalar is a quantity that has only magnitude Example: the number of passengers on the train is 160 Example: the motion of a train is 160 km/hr in a westerly direction

  4. A vector is said to be COLLINEAR to another vector if it is parallel to another vector. Blue and green and purple vectors are COLINEAR…even though they may not be pointing in the same direction

  5. Two vectors are said to be OPPOSITE to each other if they have the same NORM but OPPOSITE direction.

  6. A vector can be named using the two letters that make up its end points: Or it is simply named with a single letter with an arrow over top. Often the letters used are u, v, and w.

  7. The NORM of the vector is denoted by this notation Example: Find

  8. Two vectors that are perpendicular to each other are orthogonal. Blue and purple vectors are ORTHOGONAL

  9. How can we find the magnitude if we have the initial point and the terminal point? Q The distance formula Terminal Point magnitude is the length direction is this angle Initial Point P How can we find the direction? (Is this all looking familiar? You can make a right triangle and use trig to get the angle!)

  10. The direction is described much like using a protractor Start at the horizontal half-line directed eastward and rotate counter-clockwise As an example, the notation for this is

  11. For the orientation of the vector PQ…

  12. A vector’s direction can also be related to CARDINAL POINTS (what’s on a map!) Ex. 30 degrees West of South (S 30o W) 60 degrees West of North (N 60o W) 7 degrees East of North (N 7o E)

  13. Vectors are equal as long as the direction and magnitude are the same. It does not matter where they are located. It is easiest to find a vector with initial point at the origin and terminal point (x, y). Q Terminal Point direction is this angle Initial Point P

  14. OPERATIONS ON VECTORS You can add and subtract vectors!

  15. To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words). To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words). To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words). Terminal point of w Move w over keeping the magnitude and direction the same. Initial point of v

  16. You subtract vectors by ADDING THE OPPOSITE (or the negative) of the vector The negative of a vector is just a vector going the opposite way.

  17. Examples of subtracting vectors…just add the opposite

  18. A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

  19. A zero vector has no magnitude. It is denoted by the following notation: or The zero vector has every direction

  20. Using the vectors shown, find the following:

  21. ALGEBRAIC VECTORS

  22. CHASLES’ RELATION

  23. Homework: Page 271 #2 Page 272 #3,4 Page 274 #6,8,10 Page 280 #7,10,11

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