1.3 Vectors and Scalars

1. 1.3 Vectors and Scalars Scalar: shows magnitude Vector: shows magnitude and direction

2. 1.3 Vectors and Scalars • SCALAR • Distance • Speed • Temperature • Mass • Energy • Work • Pressure • VECTOR • Displacement • Velocity • Acceleration • Weight • Forces • Momentum • Field strengths

3. 1.3 Vectors and Scalars Resultant RESULTANT VECTOR: the sum of two or more vectors.

4. 1.3 Vectors and Scalars • Vectors can be added in any order. • To subtract a vector, add its opposite. • Multiplying or dividing vectors by scalars results in vectors.

5. 1.3 Adding Vectors • When adding vectors, their directions must be taken into account • Units must be the same • Graphical Methods • Use scale drawings • Algebraic Methods • More convenient

6. 1.3 Graphically Adding Vectors Drawing the vectors “tip-to-tail” The resultant is drawn from the origin of A to the end of the last vector Measure the length of R and its angle Use the scale factor to convert length to actual magnitude

7. 1.3 Graphically Adding Vectors When you have many vectors, just keep repeating the process The resultant is still from the origin of the first vector to the end of the last vector

8. 1.3 Alternative Graphical Method When you have only two vectors, you may use the Parallelogram Method All vectors, including the resultant, are drawn from a common origin The remaining sides of the parallelogram are sketched to determine the diagonal, R

9. 1.3 Vector Addition Vectors obey the Commutative Law of Addition The order in which the vectors are added doesn’t affect the result

10. 1.3 Vector Subtraction Special case of vector addition If A – B, then use A+(-B) Continue with standard vector addition procedure

11. 1.3 ×/÷a Vector by a Scalar If the scalar is positive, the direction of the result is the same as of the original vector If the scalar is negative, the direction of the result is opposite that of the original vector

12. 1.3 Vectors and Scalars v 2v Multiply vector by a scalar:

13. 1.3 Vectors The Pythagorean Theorem Use the Pythagorean theorem to find the magnitude of the resultant vector. c2 = a2 + b2

14. 1.3 Components of a Vector A component is a part It is useful to use rectangular components These are the projections of the vector along the x- and y-axes

15. The sine and cosine functions can be used to find the components of a vector. The sine and cosine functions are defined in terms of the lengths of the sides of right triangles.

16. Components of a Vector, cont. The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis Then,

17. 1.3 Components of a Vector • The previous equations are valid only if θ is measured with respect to the x-axis • The components can be positive or negative and will have the same units as the original vector • The components are the “legs” of the right triangle whose hypotenuse is A:

18. 1.3 Adding Vectors Algebraically • Choose a coordinate system and sketch the vectors • Find the x- and y-components of all the vectors • Add all the x and y components: • This gives Rx: • This gives Ry:

19. 1.3 Adding Vectors Not Perpendicular Chapter 3 • Suppose that a plane travels first 5 km at an angle of 35°, then climbs at 10° for 22 km, as shown below. How can you find the total displacement? • Because the original displacement vectors do not form a right triangle, you can not directly apply the tangent function or the Pythagorean theorem. d2 d1

20. 1.3 Adding Vectors Not Perpendicular Chapter 3 • You can find the magnitude and the direction of the resultant by resolving each of the plane’s displacement vectors into its x and y components. • Then the components along each axis can be added together. As shown in the figure, these sums will be the two perpendicular components of the resultant, d. The resultant’s magnitude can then be found by using the Pythagorean theorem, and its direction can be found by using the inverse tangent function.