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Lecture 2 The Vectors

Lecture 2 The Vectors. The Vectors. 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. Q. Terminal Point. magnitude is the length.

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Lecture 2 The Vectors

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  1. Lecture 2The Vectors

  2. The Vectors • A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The lengthof the vector represents the magnitude and the arrow indicates the direction of the vector.

  3. Q Terminal Point magnitude is the length direction is this angle Initial Point P

  4. 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).

  5. Vector Components • Vectors can be related to the more familiar Cartesian coordinates (x, y) of a point P in a plane: suppose P is reached from the origin by a displacement r. • Then r can be written as the sum of successive displacements in the x-and y-directions: • These are called the components of r.

  6. Define î , ĵ to be vectors of unit length parallel to the x, y axes respectively. The components are x î ,y ĵ

  7. How r Relates to (x, y) • The length (magnitude) of r is: • The angle between the vector and the x-axis is given by:

  8. This is the notation for a position vector. This means the point (a, b) is the terminal point and the initial point is the origin. Vectors are denoted with bold letters (a, b) (3, 2)

  9. If we want to find the unit vector having the same direction as a given vector, we find the magnitude of the vector and divide the vector by that value.

  10. When we want to know the magnitude of the vector (remember this is the length) we denote it Let's look at this geometrically:

  11. If we know the magnitude and direction of the vector, let's see if we can express the vector in ai + bj form. As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction.

  12. 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

  13. The negative of a vector is just a vector going the opposite way. A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

  14. Using the vectors shown, find the following:

  15. Example1: An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane? N 180o 0o 270o Scale: 50 mph = 1 inch

  16. An airplane is flying 200mph at 50o N of E. Wind velocity is 50 mph due S. What is the velocity of the plane? N W E S Scale: 50 mph = 1 inch

  17. An airplane is flying 200mph at 50o N of E. Wind velocity is 50 mph due S. What is the velocity of the plane? N 50 mph 200 mph W E VR = 165 mph @ 40° N of E S Scale: 50 mph = 1 inch

  18. VR=? V2 V1 Example 2: 24.99 N 1) Find the resultant Magnitude:__________ of the two vectors Direction:___________ Vector #1 = 20.5 N West Vector #2 = 14.3 N North Vector Diagram 34.90º N of W

  19. - + = A B C R Vector A 30 METERS @ 45O Vector B 50 METERS @ 0O + (- ) + = A B C R Vector C 30 METERS @ 90O SUBTRACTING VECTORS USING COMPONENTS Vector A 30 METERS @ 45O - Vector B 50 METERS @ 180O Vector C 30 METERS @ 90O

  20. RESOLVE EACH INTO • X AND Y COMPONENTS • AX = 30 METERS x COS 450 = 21.2 METERS • AY = 30METERS x SIN 450 = 21.2 METERS • BX = 50 METERS x COS 1800 = - 50 METERS • BY = 50METERS x SIN 1800 = 0 METERS • CX = 30 METERS x COS 900 = 0 METERS • CY = 30METERS x SIN 900 = 30 METERS

  21. (2) ADD THE X COMPONENTS OF EACH VECTOR ADD THE Y COMPONENTS OF EACH VECTOR  X = SUM OF THE Xs = 21.2 + (-50) + 0 = -28.8  Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2 (3) CONSTUCT A NEW RIGHT TRIANGLE USING THE  X AS THE BASE AND  Y AS THE OPPOSITE SIDE  Y = +51.2  X = -28.8 THE HYPOTENUSE IS THE RESULTANT VECTOR

  22.  Y = +51.2  X = -28.8 (-28.8)2 + (+51.2)2 = 58.7 (4) FIND THE LENGTH (MAGNITUDE) OF THE RESULTANT VECTOR ANGLE TAN-1 (51.2/-28.8) ANGLE = -60.6 0 (1800 –60.60 ) = 119.40 QUADRANT II (5) FIND THE ANGLE (DIRECTION) USING INVERSE TANGENT OF THE OPPOSITE SIDE OVER THE ADJACENT SIDE RESULTANT = 58.7 METERS @ 119.4O

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