Chapter 3 Two-Dimensional Motion & Vectors Hanan Anabusi

3-2 Vector Operations Objectives: Identify appropriate coordinate systems for solving problems with vectors. Apply the Pythagorean theorem and tangent function to calculate the magnitude and direction of a resultant vector. Resolve vectors into components using the sine and cosine functions. Add vectors that are not perpendicular.

3-2 Vocabulary Coordinate system Resultant Vector components

3-2 Vector operations Coordinate systems in two dimensions: the y-axis is depicted as the north/south direction and the x-axis as the east/west direction. Using mathematical instead of graphical approaches: For any right angle vectors, the resultant can be determined using the Pythagorean theorem: To determine the angle of the resultant, use tangent function: where is the angle of the resultant from the beginning point

Sample problem A An archaeologist climbs the Great Pyramid in Giza, Egypt. If the pyramid’s height is 136 m and its width is 2.30 x 102 m, what is the magnitude and the direction of the archaeologist’s displacement while climbing form the bottom of the pyramid to the top?

Practice A on page 89, questions 2 and 4. Classwork Practice A on page 89, questions 2 and 4. Answers: 2. 4.

Resolving Vectors into Components The projections of a vector along the axes of a coordinate system are called components. The x component is parallel to the x-axis, the y component is parallel to the y-axis. They can be positive or negative numbers with units. Any vector can be described by a set of perpendicular components. Resolving a vector into components can make it easier to describe directions. When resolving a vector into perpendicular components, the properties of trigonometry can be used for calculations.

Sine & Cosine functions for right triangles Sine of an angle is the ratio of the length of the side opposite that angle to the length of the hypotenuse (opposite the right angle) Cosine of an angle is the ratio of the length of the side adjacent to that angle to the length of the hypotenuse.

Sample problem B Find the component velocities of a helicopter traveling 95 km/h at an angle of 35° to the ground. use Pythagorean theorem to check magnitude

Adding vectors that are not perpendicular When vectors do not form a right triangle, the magnitude and direction of each vector’s x and y components must be determined and then added together. When the x and y components are added together, then Pythagorean theorem and tangent functions can be used to find the magnitude and direction of the resultant.

Sample problem C A hiker walks 27.0 km from her base camp at The next day, she walks 41.0 km in a direction and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.

Sample problem C (2) North y Ranger’s tower Base camp East West x South

North Sample problem C (3) West East South y y x x

Sample problem C (4)

Sample problem C (5)

Practice C on page 94, questions 2 and 4. Answers: 2. 4. Classwork Practice C on page 94, questions 2 and 4.