Lesson 10/20

Lesson 10/20

Vector Components

Open Ended

Chop Wood Carry Water

Practice Problems

Adding Vectors in 2-Dimension Practice Problems

Vector Components

• What is the direction of the vector shown in the image?

• The image isn't pointing directly up or directly to the right, but somewhere between.

• To determine the exact location we need to choose a coordinate system.

Vector Components

• The coordinated system is similar to laying a grid drawn on a piece of transparent plastic on top of a vector problem.

• You choose where to put the center of the grid (the origin) and establish the direction in which the axes point.

• Notice the x-axis is drawn through the origin with an arrow pointing in the positive direction.

• The positive y-axis is located 90 degrees from the positive x-axis and crosses the x-axis at the origin.

Vector components

• How do you choose the direction of the x-axis?

• There is never a simple answer to this.

• It is often easier to have the x-axis pointing east and the y-axis pointing north.

• If it is on an incline, it is often convenient to place the x-axis parallel to the surface of the incline.

Component Vector

• Defininging a component system allows you to describe a vector in a very useful way.

• Vector A in the image can now be described as going 5 units in the positive x-direction and 4 units in the positive y-direction.

• You can represent this information in the form of two vectors, A_x and A_y.

Component Vectors

• Notice that A_x is parallel to the x-axis and A_y is parallel to the y-axis

• If you add A_x and A_y the resultant is the original vector, A.

• A vector can be broken into its components, or a vector parallel to the x-axis and another parallel to the y-axis.

• A= A_x + A_y

Components of Vectors

• The process of breaking a vector into its components is called vector resolution.

• When we sketch it out, the original vector is the hypotenuse of a right triangle.

• This means that the magnitude of the original vector will always be greater than or equal to the magnitude of either of the component vectors.

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Direction in a component system

• In a two-dimensional force problem, carefully select your coordinate system.

• The direction of any vector will be specified relative to those coordinates.

• We define the direction of a vector as the angle that the vector makes with the x-axis.

Directions in a coordinate system

• In addition to measuring the component vector grahically, you can find the components using trigonometry.

• Ax = cos ( $\theta$  )

• Ay= sin ( $\theta$  )

Open Ended

Exit Ticket - If A_x is negative, which two quadrants could it possibly be in?