1. vectors

2. Vectors and Direction Vectors are quantities that have a size and a direction. Vectors are quantities that have a size and a direction. A quantity is something that you measure. A quantity is something that you measure. Some quantities have only size. Some quantities have only size. Some quantities have both a size and a direction. Some quantities have both a size and a direction. Vectors are symbolized with arrows. Vectors are symbolized with arrows.

3. The difference between scalars and vectors The difference between scalars and vectors If a quantity has only a size, it is called a scalar. Time and temperature are examples of scalars. If a quantity has only a size, it is called a scalar. Time and temperature are examples of scalars. If a quantity has a size and a direction, it is called a vector and can be symbolized, or drawn, as an arrow. Velocity is an example of a vector. If a quantity has a size and a direction, it is called a vector and can be symbolized, or drawn, as an arrow. Velocity is an example of a vector.

4. The properties of vector You show a vector by bold font or an arrow over its symbol. You show a vector by bold font or an arrow over its symbol. You can move a vector parallel to itself. The vector from (2,5) to (6,-2) is the same as the one from the origin to (4,-7) You can move a vector parallel to itself. The vector from (2,5) to (6,-2) is the same as the one from the origin to (4,-7) Multiplying a vector by a positive scalar does not change the direction of the vector. Multiplying a vector by a positive scalar does not change the direction of the vector.

5. Multiplying a vector by a negative scalar results in the vector going the opposite direction Multiplying a vector by a negative scalar results in the vector going the opposite direction A unit vector is a vector with magnitude one. We may easily convert any vector into a unit vector simply by dividing each component by the magnitude of the vector A unit vector is a vector with magnitude one. We may easily convert any vector into a unit vector simply by dividing each component by the magnitude of the vector A vector can be written in terms of a rectangular coordinate Vector v, from (0, 0) to (a, b) is represented as: v =ai+bj where a is the A vector can be written in terms of a rectangular coordinate Vector v, from (0, 0) to (a, b) is represented as: v =ai+bj where a is the horizontal component of v, and b is the vertical component of v. horizontal component of v, and b is the vertical component of v. The magnitude of v is The magnitude of v is

6. The direction angle of v is the angle that v makes with the positive x-axis The direction angle of v is the angle that v makes with the positive x-axis When you want to add or subtract vectors, first, you add or subtract the horizontal and then you add or subtract the vertical components When you want to add or subtract vectors, first, you add or subtract the horizontal and then you add or subtract the vertical components When you multiply a vector by a scalar, each component is multiplied by the scalar. When you multiply a vector by a scalar, each component is multiplied by the scalar. Two vectors are equal only if their magnitudes are equal and their directions are equal Two vectors are equal only if their magnitudes are equal and their directions are equal

7. The dot product of two vectors gives a scalar answer. To find the dot product, you must know the length of each vector and the angle between them (θ): The dot product of two vectors gives a scalar answer. To find the dot product, you must know the length of each vector and the angle between them (θ): A.B = AB cosθ A.B = AB cosθ For example, work is a scalar product of the force vector and the distance vector. For example, work is a scalar product of the force vector and the distance vector. W = F. d = Fdcosθ

8. The cross product of two vectors gives a vector answer. The direction of that answer is perpendicular to the plane that contains A & B. The length of the answer comes from: The cross product of two vectors gives a vector answer. The direction of that answer is perpendicular to the plane that contains A & B. The length of the answer comes from: |A x B| = AB sinθ |A x B| = AB sinθ

9. the direction AxB Point your index finger in the direction of the first vector. Cross your middle finger under your index finger. Your middle finger represents the second vector. Your thumb is then the resultant. Point your index finger in the direction of the first vector. Cross your middle finger under your index finger. Your middle finger represents the second vector. Your thumb is then the resultant.

10. Right hand rule

11. Additional Exercis u = 3i - and v = -2i + 5j, find u = 3i - and v = -2i + 5j, find

12. Find the length and direction for the displacement vector from (3.0 meters,4.0 m) to (-1.0 m,-2.0 m). Find the length and direction for the displacement vector from (3.0 meters,4.0 m) to (-1.0 m,-2.0 m).

13. Find the x and y components of the momentum, p = 5.0 kg m/s at 130 o. Find the x and y components of the momentum, p = 5.0 kg m/s at 130 o.

14. A force of 8 Newtons is applied 30 o above the horizontal. In the triangle, the cosine of 30 o is x/8.So the x- component is (8N) x cos 30 o. Similarly, the y-component is (8N) x sin 30 o A force of 8 Newtons is applied 30 o above the horizontal. In the triangle, the cosine of 30 o is x/8.So the x- component is (8N) x cos 30 o. Similarly, the y-component is (8N) x sin 30 o

15. the vector coordinates are (3.0,4.0): the length (by Pythagorean Theorem) is the square root of (3 2 + 4 2 ) = 5.0. the vector coordinates are (3.0,4.0): the length (by Pythagorean Theorem) is the square root of (3 2 + 4 2 ) = 5.0. The direction is determined by the tangent of the angle = (4.0/3.0). The direction is determined by the tangent of the angle = (4.0/3.0).