1. Some Key Concepts Scalars and Vectors Multiplying Scalars with Vectors
Vector and Scalar components
Components of Vectors
Resultant ( or Vector Sum)
Equilibrant
Displacement Vector

2. Vectors and Scalars A vector has magnitude as well as direction.
Some vector quantities: displacement, velocity, force, momentum
A scalar has only a magnitude.
Some scalar quantities: mass, time, temperature
Figure 3-1. Caption: Car traveling on a road, slowing down to round the curve. The green arrows represent the velocity vector at each position.

3. Addition of Vectors—Graphical Methods
For vectors in one dimension, simple addition and subtraction are all that is needed.
You do need to be careful about the signs, as the figure indicates.
Figure 3-2. Caption: Combining vectors in one dimension.

4. Addition of Vectors—Graphical Methods
If the motion is in two dimensions, the situation is somewhat more complicated.
Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem.
Figure 3-3. Caption: A person walks 10.0 km east and then 5.0 km north. These two displacements are represented by the vectors D1 and D2, which are shown as arrows. The resultant displacement vector, DR, which is the vector sum of D1 and D2, is also shown. Measurement on the graph with ruler and protractor shows that DR has a magnitude of 11.2 km and points at an angle θ = 27° north of east.

5. Adding the vectors in the opposite order gives the same result:
Figure 3-4. Caption: If the vectors are added in reverse order, the resultant is the same. (Compare to Fig. 3–3.)

6. Addition of Vectors—Graphical Methods
Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.
Figure 3-5. Caption: The resultant of three vectors: vR = v1 + v2 + v3.

7. Addition of Vectors—Graphical Methods
The parallelogram method may also be used; here again the vectors must be tail-to-tip.
Figure 3-6. Caption: Vector addition by two different methods, (a) and (b). Part (c) is incorrect.

8. Subtraction of Vectors, and Multiplication of a Vector by a Scalar
In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction.
Then we add the negative vector.
Figure 3-7. Caption: The negative of a vector is a vector having the same length but opposite direction.
Figure 3-8. Caption: Subtracting two vectors: v2 – v1.

9. Multiplication of a Vector by a Scalar
A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.
Figure 3-9. Caption: Multiplying a vector v by a scalar c gives a vector whose magnitude is c times greater and in the same direction as v (or opposite direction if c is negative).

10. Adding & Subtracting Vectors
Vectors can be added or subtracted from each other graphically.
Each vector is represented by an arrow with a length that is proportional to the magnitude of the vector.
Each vector has a direction associated with it.
When two or more vectors are added or subtracted, the answer is called the resultant.
A resultant that is equal in magnitude and opposite in direction is also known as an equilibrant.

11. Adding Vectors using the Pythagorean Theorem
If the vectors occur such that they are perpendicular to one another, the Pythagorean theorem may be used to determine the resultant. =
3 m
4 m
5 m +
4 m
3 m
A B = C2
(4m) (3m) = (5m)2
When adding vectors, place the tail of the second vector at the tip of the first vector.

12. Adding & Subtracting Vectors
If the vectors occur in a single dimension, just add or subtract them. + =
4 m
7 m
3 m
7 m + - =
4 m
7 m
3 m
7 m
When adding vectors, place the tail of the second vector at the tip of the first vector.
When subtracting vectors, invert the second one before placing its tail at the tip of the first vector.

13. Adding Vectors by Components
Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.
Figure Caption: Resolving a vector V into its components along an arbitrarily chosen set of x and y axes. The components, once found, themselves represent the vector. That is, the components contain as much information as the vector itself.

14. 3-4 Adding Vectors by Components
Remember:
soh
cah
toa
Figure Caption: Finding the components of a vector using trigonometric functions.
If the components are perpendicular, they can be found using trigonometric functions.

15. Adding Vectors by Components
The components are effectively one-dimensional, so they can be added arithmetically.
Figure Caption: The components of v = v1 + v2 are
vx = v1x + v2x
vy = v1y + v2y

16. Adding Vectors by Components
Draw a diagram; add the vectors graphically.
Choose x and y axes.
Resolve each vector into x and y components.
Calculate each component using sines and cosines.
Add the components in each direction.
To find the length and direction of the vector, use:
and .

17. Adding Vectors by Components
Three short trips.
An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45°) for 440 km; and the third leg is at 53° south of west, for 550 km, as shown. What is the plane’s total displacement?
Figure 3-14.
Answer: The x and y components of the displacement are 600 km and -750 km.
Alternatively, the magnitude of the displacement is 960 km, at an angle of 51° below the x axis.

18. Algebraic Addition
In the event that there is more than one vector, the x-components can be added together, as can the y-components to determine the resultant vector. y R
Rx = ax + bx + cx
Ry = ay + by + cy
R = Rx + Ry cx cy a b c bx by ay ax x

19. Properties of Vectors
A vector can be moved anywhere in a plane as long as the magnitude and direction are not changed.
Two vectors are equal if they have the same magnitude and direction.
Vectors are concurrent when they act on a point simultaneously.
A vector multiplied by a scalar will result in a vector with the same direction. P
F = ma
vector
scalar
vector

20. Properties of Vectors (cont.)
Two or more vectors can be added together to form a resultant. The resultant is a single vector that replaces the other vectors.
The maximum value for a resultant vector occurs when the angle between them is 0°.
The minimum value for a resultant vector occurs when the angle between the two vectors is 180°.
The equilibrant is a vector with the same magnitude but opposite in direction to the resultant vector.