Vector Operations Unit 2.3

Coordinate Systems in 2 D Two methods can be used to describe motion: one axis {x-axis} two axis {x-axis; y-axis}

Coordinate Systems in 2 D 1 dimension y v = 300 m/s Northeast

Coordinate Systems in 2 D The problem with this method is that the axis must be turned again if the direction of the object changes. It will also become difficult to describe the direction of another object if it is not traveling exactly Northeast.

Coordinate Systems in 2 D The addition of another axis not only helps describe motion in two dimensions but also simplifies analysis of motion in one dimension. v = 300 m/s Northeast

Coordinate Systems in 2 D When analyzing motion of objects thrown into the air orienting the y-axis perpendicular to the ground and therefore parallel to the direction of the free fall acceleration simplifies things.

Coordinate Systems in 2 D There are no firm rules for applying to the coordinate system. As long as you are consistent, your final answer should be correct. This is why picking a frame of reference is extremely important.

Vector Resultant In order to determine the resultant magnitude and direction, we can use two different methods: 1) Pythagorean Theorem 2) Tangent Function

c2 = a2 + b2 Vector Resultant The Pythagorean theorem states that for any right angle, the square of the hypotenuse (side opposite to the right angle) equals the square of the other 2 sides. c2 = a2 + b2

Vector Resultant Use the tangent function to find the direction of the resultant. Opposite hypotenuse adjacent tanѲ = opposite = ∆ y adjacent ∆ x

Resolving Vectors in Components

Vector Components The horizontal and vertical parts that add up to give displacement are called components. The x-component is parallel to the x-axis. The y-component is parallel to the y-axis.

Vector Components The components can be either positive or negative. Any vector can be completely described by a set of perpendicular components by breaking a single vector into two components or resolving it into its components.

Vector Components An object’s motion can sometimes be described more conveniently in terms of directions such as north to south or east to west.

Vector Components A key to solving problems of motion is to recognize that a right angle can be drawn using velocity and its x & y components. The situation can then be analyzed using trigonometry.

Vector Components The sine and cosine functions are defined in terms of the length of side of such right triangle.

Vector Components sin θ = opposite hypotenuse hypotenuse opposite cos θ = adjacent adjacent hypotenuse

Vector Components We have been dealing with vectors that are perpendicular to one another. In real life, many objects move in one direction and then turn at an acute angle before continuing their motion.

Vector Components Because the original displacement vectors do not form a right triangle, it is not possible to directly apply the tangent function or the Pythagorean theorem when adding the two original vectors.

Vector Components Determining the magnitude and the direction of the resultant can be achieved by resolving each of the plane’s displacement vectors into their x and y components. Then the components along each axis can be added together.

Vector Components

Vector Components