VECTORS AND BEARING for Junior High school.

Vectors and bearing for trainees

1. VECTORS AND BEARING FAUZIA GROUP

2. What is a vector? Vector is a physical quantity that has both direction and magnitude. In other words, the vectors are defined as an object comprising both magnitude and direction. It describes the movement of the object from one point to another. The below figure shows the vector with head, tail, magnitude and direction.

3. COMPONENTS OF VECTOR Magnitude D irection

4. MAGNITUDE T he magnitude of a vector is the length of the vector. T he magnitude of the vector a is denoted as I t is square root of the sum of squares of the vector.

5. DIRECTION The direction of a vector is the orientation of the vector, that is the angle it makes with the x-axis. A vector is drawn by a line with an arrow on the top and a fixed point at the other end.

6. TYPES OF VERTORS Zero Vector Unit Vector Position Vector Co-initial Vector Like and Unlike Vectors

7. TYPES CONT. Co-planar Vector Collinear Vector Equal Vector Displacement Vector Negative of a Vector

8. Zero Vector A zero vector is a vector when the magnitude of the vector is zero and the starting point of the vector coincides with the terminal point. In other words, for a vector the coordinates of the point A are the same as that of the point B then the vector is said to be a zero vector and is denoted by 0. This follows that the magnitude of the zero vector is zero and the direction of such a vector is indeterminate.

9. Unit Vector A vector which has a magnitude of unit length is called a unit vector. Suppose if is a vector having a magnitude x then the unit vector is denoted by x ̂ in the direction of the vector and has the magnitude equal to 1.

10. Position Vector If O is taken as reference origin and P is an arbitrary point in space then the vector is called as the position vector of the point. Position vector simply denotes the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin.

11. Co-initial Vectors The vectors which have the same starting point are called co-initial vectors. The vector and are called co-initial vectors as they have the same starting point

12. Like and Unlike Vectors The vectors having the same direction are known as like vectors. On the contrary, the vectors having the opposite direction with respect to each other are termed to be unlike vectors.

13. CO-PLANAR VECTORS Three or more vectors lying in the same plane or parallel to the same plane are known as co-planar vectors.

14. Collinear Vectors Vectors that lie along the same line or parallel lines are known to be collinear vectors. They are also known as parallel vectors. Two vectors are collinear if they are parallel to the same line irrespective of their magnitudes and direction. Thus, we can consider any two vectors as collinear vectors if and only if these two vectors are either along the same line or these vectors are parallel to each other in the same direction or opposite direction. For any two vectors to be parallel to one another, the condition is that one of the vectors should be a scalar multiple of another vector. The below figure shows the collinear vectors in the opposite direction.

15. Equal Vectors Two or more vectors are said to be equal when their magnitude is equal and also their direction is the same. The two vectors shown above, are equal vectors as they have both direction and magnitude equal.

16. Displacement Vector If a point is displaced from position A to B then the displacement AB represents a vector which is known as the displacement vector .

17. Negative of a Vector If two vectors are the same in magnitude but exactly opposite in direction then both the vectors are negative of each other. Assume there are two vectors a and b, such that these vectors are exactly the same in magnitude but opposite in direction then these vectors can be given by a = – b

18. Operations of Vectors Addition of vectors Subtraction of vectors Multiplication of Vectors

19. Addition of Vectors Let us consider there are two vectors P and Q, then the sum of these two vectors can be performed when the tail of vector Q meets with the head of vector A. And during this addition, the magnitude and direction of the vectors should not change. The vector addition follows two important laws, which are; Commutative Law: P + Q = Q + P Associative Law: P + (Q + R) = (P + Q) + R

20. Subtraction Of Vectors Here, the direction of other vectors is reversed and then the addition is performed on both the given vectors. If P and Q are the vectors, for which the subtraction method has to be performed, then we invert the direction of another vector say for Q, make it -Q. Now, we need to add vector P and -Q. Thus, the direction of the vectors are opposite each other, but the magnitude remains the same. P – Q = P + (-Q)

21. MULTIPLICATION OF VECTORS If k is a scalar quantity and it is multiplied by a vector A, then the scalar multiplication is given by kA. If k is positive then the direction of the vector kA is the same as vector A, but if the value of k is negative, then the direction of vector kA will be opposite to the direction of vector A. And the magnitude of the vector kA is given by |kA|.

22. Bearings In mathematics, a bearing is the angle in degrees measured clockwise from north. Bearings are usually given as a three-figure bearing. For example, 30° clockwise from north is usually written as 030°.

23. KINDS OF BEARING C ompass bearing is based on the four main cardinal points. (North, East, South, West) T hree figures bearing: A n angle measured in degrees M easured as angle in the clockwise direction from the geographic North T he angles are given in 3 figures or digits from 000 ˚ to 360˚

24. E XAMPLE A n aeroplane flew from town A, travels 200km at a bearing of 135 ˚to town B. I t then travels 250km from B at a bearing of 045˚ to town C. Find: T he distance of A from C T he bearing of A from C

25. Back or Reverse bearing This is a bearing in surveying resulting from a backsight. Formula ^ + 180° ( if angle is less than 180° ) Or ^ - 180° ( if angle is greater or more than 180° )