Vector Diagrams Motion in Two Dimensions

Learning Targets Describe the characteristics of a vector diagram Create vector diagrams for perpendicular vectors Calculate the magnitude and direction of a resultant vector

What are Vector Diagrams? Vectors can be represented graphically using scaled vector diagrams. In these diagrams, vectors are represented by arrows that point in the direction of the vector. The length of the vector arrow is proportional to the vector’s magnitude

Characteristics of Vector Diagrams A scale is clearly listed A vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail. The magnitude and direction of the vector is clearly labeled.

Direction of a Vector The direction of a vector can be expressed as a counterclockwise angle of rotation of the vector about its “tail" from due East. A vector with a direction of 240 ° means that if the tail of the vector was pinned down, the vector would be rotated 240 ° counterclockwise from due east.

The direction of a vector can also be expressed as an angle of rotation from a specific direction For example, a vector can be said to have a direction of 40 degrees North of East This means the vector pointing East has been rotated 40 degrees towards the northerly direction

Magnitude of a Vector The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow.

Resultant Vectors The displacement from the tail of the first vector to the head of the last vector is called the resultant

Determining the Resultant For vectors that are perpendicular to one another, the Pythagorean theorem and the inverse tangent function can be used to determine the magnitude and direction of the resultant vector

The Pythagorean Theorem The Pythagorean theorem can be used to find the magnitude of the resultant (hypotenuse) if you know the magnitude of both the x and y components Hypotenuse2 = Length leg one2 + Length of leg two2 Resultant

Angle = tan-1 (opposite leg (y) / adjacent leg (x)) The Tangent Function The inverse tangent function can be used to find the direction of the resultant For any right triangle, the tangent of an angle is defined as the ratio of the opposite and adjacent legs Angle = tan-1 (opposite leg (y) / adjacent leg (x))

Sample Problem A squirrel trying to get down a tree travels 2.5 m east across a branch and then 17 m down the tree. What is the magnitude and direction of the squirrel’s displacement? 24 m 81.6˚ S or E